The Hillsborough departed Portsmouth in December 1798.
The Hillsborough had the greatest mortality rate amongst the convicts as had ever been seen. Later an enquiry to investigate the circumstances and correspondence to the Transport Office revealed that prisoners were already ill before the Hillsborough even weighed anchor.
Gaol Fever in the Hulks
Following is an extract from the Annals of Medicine outlining the methods used in attempt to stem the gaol fever sweeping through the prisons and hulks in 1798:
We are here presented with several communications from Mr Samuel Hill, surgeon at Portsea, who had the charge of the hulks in Langstone harbour, near that town. A fever of a contagious nature made its appearance on board these hulks, in the month of July 1798, which soon became alarming. The number of patients continued increasing from the 6th of July to the 29th of August.
In the former month, sixteen were attacked; in the latter sixty-six. Upon Mr Hill's representing to Mr Dyne, contractor for the care of the convicts, that he thought benefit would ensue, if the method of fumigating recommended by Dr Smyth was put in practice in Langstone harbour, he, with great humanity, ordered every necessary article, and gave directions that no expense might be spared in attempting to stop the progress of the fever. The fumigation with nitrous vapour was accordingly begun on board the Sincerity hospital-ship, on the 29th of August 1798; and it was with extreme pleasure that Mr Hill soon observed the good effects of this practice on many of the fever-patients.
Finding the most beneficial effects from fumigation on board the hospital-ship, in bringing the fever sooner to a conclusion, by shortening all its stages, Mr Hill resolved to apply it also to the source of the contagion; and accordingly the prison-hulks, La Fortune and Ceres, were fumigated every night, from October 15. to November 20. Mr Hill had soon the pleasure of finding the number of the sick men reduced ; and seven days having elapsed without one patient having been sent to the hospital, the fumigation was discontinued.
But on the 21st of November, eight men were received from the Hillsborough, Botany Bay ship, one of which number was in the last stage of a contagious fever, and two laboured under dysentery. Several patients in a state of recovery, caught the new contagion; and many attendants were taken ill with diarrhoea and dysentery.
(Eleven more men were sent to the hospital on 30th November and five on the 19th December )
As it was impossible to prevent communication with the prison-hulks, the prisoners again became sickly, and many died, some of whom were not ill three days before that event took place. The fumigation was again resorted to on the 26th of November, and continued to the time when Mr Hill's account was written, January 13. 1799. On this occasion, the good effects derived from it were equally conspicuous as before, and Mr Hill had the satisfaction of saying, that there had not been a patient received for the last eighteen days, and that there was not a single fever-patient in the hospital.
Missionaries for the Cape
Dr. Vanderkemp, accompanied by three brother missionaries, Messrs. Kitcherer, Edwards, and Edmonds, sailed for the Cape of Good Hope on the Hillsborough.......
Among these miserable creatures the missionary brethren determined to commence a course of instruction. They were told, indeed, that if they ventured into the hold among the convicts, they would certainly throw a blanket over them, and rob them of whatever they had in their pockets; but, notwithstanding this representation, the missionaries determined to make the attempt, and happily they were received with every mark of respect, and listened to with the greatest attention. By the kindness and affability of their manners, they in a few days so conciliated the regard of the prisoners, that they found themselves completely at their ease among them, ventured into the midst of them without the smallest dread, and conversed as freely with them as if they had been their most intimate friends and acquaintances.
This was the more remarkable, considering the manner in which others were handled by them. One day, before they sailed from Portsmouth, several naval officers came on board in search of some deserters, who, it was supposed, had concealed themselves among the convicts; but no sooner had one of the officers, with his men, attempted to pass the entrance of the orlop deck, than the prisoners seized him, beat him most unmercifully, and wounded him in the head with his own dagger. Two days after, a cutter, with some officers, and a detachment of marines, came to renew the search; but the convicts threatening to murder them if they entered the hold, they wisely desisted from the attempt.
About the same time, the prisoners engaged in a plot to murder the officers of the Hillsborough, seize the vessel, and carry her over to France; and, though the conspiracy was providentially discovered and defeated, yet this did not hinder them, about ten days after, from entertaining the horrid design of sinking the vessel, and escaping in the boats; and, with this view, many of them had even found means to cut off their chains and handcuffs.
Such was the description of men among whom the missionaries sought to labour at the hazard of their lives. About two hundred and forty of these miserable creatures were chained in pairs, hand to hand or leg to leg, in the orlop deck, to which no light could find admission except at the hatchways. At first, the darkness of the place, the rattling of the chains, and the dreadful imprecations of the prisoners, suggested ideas of the most horrid nature, and combined to form a lively picture of the infernal regions.
Besides, in a short time, a putrid fever broke out among the convicts, and carried off no fewer than thirty-four of them during the voyage to the Cape of Good Hope. The state of the prison was now loathsome beyond description, yet, in this as well as in the hospital, surrounded with infection, disease, and death, did the missionaries daily labour to pluck these brands from everlasting burnings. Nor did they seem to labour in vain. In a short time, a number of these unhappy outcasts appeared to be impressed with anxious concern about their souls. Some of them even agreed to have a prayer meeting among themselves three times a week. Many who once could scarcely speak but to blaspheme, had learned the songs of Zion, and their horrid imprecations were changed into the language of humble praise....Chapters on Prisons and Prisoners by Joseph Kingsmill
Frank Clune in Bound for Botany Bay, quoted from William Noah's journal:
1st January 1799.
'Remainder of the convicts ordered on deck, their irons examined, and if cut some were punished with one dozen stripes, and some six dozen. We are now suffering closely from want of provisions and indeed Death would have been a welcome friend. I was not among those convicts but we all shared alike.'
2nd January 1799.
'Tranquillity was now restored. I am certain no evil intent of the convicts had been thought of, it being the intention of the convicts only to single iron themselves, but the Captain had got to such a pitch that I thought he would have hanged some of them'
The Hillsborough arrived in Port Jackson on 26 July 1799. Three hundred prisoners sailed on her and 95 of them had perished on the voyage.
Governor Hunter informed Under-Secretary King of the condition of the convicts of the Hillsborough in correspondence datd 28th July......
The Hillsborough transport, being just arriv'd in this port with a cargo of the most miserable and wretched convicts I have ever beheld, I am constrain'd to recur to my many official letters on the subject of slop cloathing and blankets. Were you, my dear sir, in the situation in which I stand, I am convinc'd all the feelings of humanity, every sensation which can occasion a pang for the distresses of a fellow creature, would be seen to operate in you with full force. Figure to yourself a ship having out of three hundred people embarked in England, and having stopped for their refreshment several weeks at the Cape Good Hope, yet having upon her voyage buried of the above number ninety five and four since landing; those who still survive are in the most sickly and wretched state, put on board the ship in England with the cloaths only in which they stood, consequently arrived here naked, where cloathing is not to be found. Nor is it possible, my dear sir, when you look back to our last general supply which was by the Sylph near three years ago and very moderate in point of quantity that you can wonder we should at this time be without. 
Captain Patton and Captain Rain's correspondence to the Transport Office in June 1800 reveal that there was already illness on board the Hillsborough before she departed England.(Extract).....
Captain (Charles?) Patton to the Transport Office.....
I beg leave to acquaint you that the Hillsborough convict ship arrived upon the Mother Bank from the Downs on the 17th November 1798. It appeared by the master's report that some of the convicts were sick, and that one convict and one child died on the passage from the river. Sir J. Fitzpatrick was requested to go on board the Hillsborough and select such of the convicts as, in his opinion, ought to be sent to the hospital ship and to give such directions as he might think necessary for the preservation of the health of those who remained on board, and that were to be embarked at this port. ....On the 20th the sick were landed and the ship preparing under the inspection of Sir J. Fitzpatrick. The logwood and other articles demanded by Sir J. Fitzpatrick were immediately ordered and all his suggestions and demands were complied with. Bedding and clothing were supplied to each convict embarked (The clothing consisted of 1 blue jacket and waistcoat, 1 pair of Russia duck trousers, 2 checked shirts, 2 pair of stockings 1 pr of shoes and 1 woollen cap.) Sir J. Fitzpatrick continued to remove the sick and replace them from the hulks to the day of the Hillsborough sailing...I beg leave to observe that the Hillsborough was a very large ship, and that no complaint was ever made to me of want of room or sufficient space for the convicts. She was fitted in the river. That the sickness did not originate from her being crowded seems evident, because it had taken place in a considerable degree on her passage from the river to Portsmouth when only half the number of convicts was embarked.....Perhaps the convicts might (in future) be allowed more air with a guard of soldiers than with a guard of seamen, who have other matters to attend to. 
Captain Rains correspondence to the Transport Office
9 June 1800
(Extract)..... I beg to acquaint you that the Hillsborough was fitted on an improved plan from any of the convict ships that preceded her, having the barrs of the prisons built so far apart as to admit of infinitely more air than could have circulated in the Barwell, which was a ship as nearly as could be of equal tonnage, embarked the same number of convicts and I am informed lost only four or five on the passage, although she had less aerial space than the Hillsborough. The 152 convicts embarked at Gravesend on board the Hillsborough were inspected as to their health by Sir J. Fitzpatrick. The usual cloathing for the voyage was supplied them; but I am of opinion there should be a double suit, as the cloaths given to them when they embark must be nearly worn out by the time they arrive at the colony. As to the miserable mattrass and one blanket which Governor Hunter speaks of it is of the same quality as those supplied to His Majestys seamen and soldiers. Since writing the preceding I have seen Sir J. Fitzpatrick, who informs me that he objected to any of the convicts at Langstone Harbour being embarked in the Hillsborough as the jail fever had raged there with much violence and he was fearful the infection remained latent in their blood; but contrary to his wishes, they were sent on board. Five of them he insisted on being returned to the hulks, all of whom died in a few days after and he says it is his firm opinion that a fever was carried out in the ship by the convicts sent from Langstone. 
Notes and Links
1). John Hardy arrived as a convict on the Hillsborough. He was executed in June 1800 and buried in the Old Sydney Burial Ground
3). William Noah was sentenced to death for burglary in 1797 at the age of 43. The sentence was commuted to transportation for life. From Newgate Prison he was taken to the Hillsborough. He received a conditional pardon in 1815 and an absolute pardon in 1818. He became a clerk in the Government lumber yard and died in 1827..Read more about the voyage in the Journal of William Noah (A Voyage to Sydney in New South Wales in 1798 1799 and A Few Remarks of the County of Cumberland in New South Wales 1798 - 1799)