Evidence suggests the cat contracted Covid-19 from its owners who had previously tested positive, the government said.
Both the cat and its owners have since made a full recovery and there was no transmission to other animals or people in the household.
Downing Street said the cat had symptoms including shortness of breath.
“Its symptoms were a respiratory infection with a nasal discharge and some shortness of breath.”
The infection was confirmed following tests at the Animal and Plant Health Agency (APHA) laboratory in Weybridge, Surrey on Wednesday July 22.
The advice from Public Health England is for people to wash their hands regularly, including before and after contact with animals.
Scientists assume pets get the virus the same way that humans do, through droplets, ITV News Science Editor Tom Clarke explains
Chief Veterinary Officer Christine Middlemiss said the infection is a "very rare event".
There is "no evidence" to suggest that pets directly transmit the virus to humans, she said.
She added how infected animals detected so far have only shown "mild clinical signs" and they usually recover "within in a few days".
Medical Director at Public Health England Yvonne Doyle said the animal's infection "should not be cause for alarm".
There have been a very small number of confirmed cases in pets in other countries in Europe, North America and Asia.
In April four tigers and three lions at the Bronx Zoo in New York tested positive for coronavirus, while weeks later two pet cats in the same state also caught the virus.
Medical Director Doyle said: "The investigation into this case suggest that the infection was spread from humans to animal, and not the other way round.
"At this time, there is no evidence that pets can transmit the disease to humans."
The cat was initially diagnosed by a private vet with feline herpes virus, a common cat respiratory infection, but was also tested for coronavirus as part of a research programme.