Alan Steele was diagnosed with the infection on the Diamond Princess, which is quarantined off Yokohama in Japan.
There are 78 people with British passports - including crew - on the ship, sources told the PA news agency.
It comes as global data shows that the number of reported new cases of coronavirus has started to decline, but it is too early to say whether this will continue.
Mr Steele, from Wolverhampton, said on Friday that he was being taken to hospital while his wife, Wendy, remained on board the ship.
Mr Steele posted on Facebook: "Just to let you all know I have been diagnosed as having the virus and am being shipped to hospital.
"Would also like to say that at the moment I am not showing any symptoms so just possibly a carrier.
"Will let you know how I am going on when possible."
Total number of cases
Princess Cruises said an additional 41 people, including Mr Steele, had tested positive for the virus on the Diamond Princess, taking the total number of cases to 61.
The company said the quarantine on the ship was due to end on February 19, barring "unforeseen developments", and confirmed that all the affected guests were being taken to hospital.
There are no plans to fly anyone off the ship and back to the UK at the moment.
A separate ship in Hong Kong, the World Dream, has about 66 British passport holders on board. Nobody on that ship has tested positive.
It comes as data from the World Health Organisation shows the reported number of new cases has fallen over the past two days.
Will number of cases fall?
Experts cautiously welcomed the decline in the reported number of new cases over the past two days.
There were roughly 3,900 new cases reported worldwide on February 5, 3,700 on February 6 and 3,200 on February 7 - the vast majority in China, World Health Organisation (WHO) figures show.
But experts warned it is not yet clear if this represents an actual fall in cases or whether the fall will prove sustainable.
The Department of Health and Social Care said that 620 people in the UK have been tested for coronavirus as of 2pm Friday, with three cases confirmed.
It is understood that the third person in the UK to be diagnosed with coronavirus caught the illness in Singapore.
He is reported to be a middle-aged British man and is understood to be the first UK national to contract the disease.
He is thought to have been diagnosed in Brighton and was transferred to St Thomas' Hospital in London, where there is an infectious disease unit, on Thursday afternoon.
Two other patients, who had recently travelled from China, are still being treated at the Royal Victoria Infirmary infectious diseases centre in Newcastle.
One is a student at the University of York, while the other is a family member.