Three more patients in England have tested positive for coronavirus, bringing the total number of UK cases to 23.
Chief medical officer Professor Chris Whitty said two of the patients had recently travelled back from Italy while the other had returned from Asia.
The cases are from Gloucestershire, Hertfordshire and Berkshire.
Prof Whitty said: "All three are being investigated and contact tracing has begun."
Downing Street said Boris Johnson had spoken with Prof Whitty and the Health Secretary Matt Hancock on Saturday afternoon to ensure everything is being done to limit the spread of the virus and that public services are prepared.
Meanwhile the first case of coronavirus in the Irish Republic has been confirmed in the east of the country.
Schools, councils and other institutions would be able to suspend some laws to cope with a pandemic.
The measures could also include allowing teachers and nursery workers to have larger classes to cope with staff absences.
Meanwhile, experts have said it is "crucial" to find out how the first person who caught the coronavirus within the UK was exposed to it, as authorities race to piece together their movements.
News of the domestic infection on Friday is a significant moment in the country's battle to stop the coronavirus.
All the other UK cases had been infected abroad.
"Contact tracing" has started on the patient.
Prof Whitty said it was not clear whether the person had contracted it "directly or indirectly" from someone who had recently returned from abroad.
The patient, who lives in Surrey, has been taken to a specialist centre at Guy's and St Thomas' in London.
Haslemere Health Centre in Surrey was closed on Friday, with a statement on its website saying: "The surgery is temporarily closed today to enable a clean of the surgery as a routine precautionary measure."
The surgery has since re-opened.
Jonathan Ball, professor of molecular virology at the University of Nottingham, said it marked a "new chapter" as there was "no known link to an affected area or known case".
"It will be crucial to understand where the infection came from to try to prevent more extensive spread," he said.
He warned the virus "can easily go under the radar" because its symptoms are very similar to a cold or mild flu.
A virus expert from the University of Leeds said the first person-to person transmission was "only a matter of time".
"What now becomes critically important is our ability to identify, isolate and care for infected individuals, and to trace their recent contacts," said Dr Stephen Griffin.
He said that small, localised outbreaks might increase in the UK in the coming weeks - and that it is vital they are contained to keep the situation manageable.
"If we experience a burgeoning epidemic as seen in South Korea it will represent a significant challenge to our already stretched NHS and public health infrastructure," Dr Griffin added.
In other coronavirus developments:
The prime minister, who will chair Monday's COBRA meeting, has been accused of being slow to act over the outbreak, but he insisted it was the government's "top priority".
He said people are "right to be concerned" about the spread of COVID-19, but the "best advice" to prevent wider spread is to wash your hands with hot water and soap.
The UK chief medical officers have raised the risk to the public from low to moderate but the risk to individuals is said to remain low.
Meanwhile, a British man in his 70s who was on board the quarantined Diamond Princess cruise ship in Japan has become the first UK fatality of the coronavirus.
The Foreign Office is advising against travel to China, South Korea, Iran and northern Italy, locations where the epidemic is most severe.
In Tenerife, hundreds of guests have been confined to the H10 Costa Adeje Palace after at least four tourists, including an Italian doctor, were diagnosed with coronavirus.
However, six Britons were among those who were told they could leave on Friday by Spanish authorities because they arrived at the hotel on Monday - after those who tested positive had been taken to hospital.
The Foreign and Commonwealth Office (FCO) has called on the Iranian government to "immediately allow" health officials into a prison in Tehran where Nazanin Zaghari-Ratcliffe believes she has contracted coronavirus.
Her husband Richard Ratcliffe had earlier raised concerns at Evin prison's "refusal to test her" for the virus, which has spread rapidly in Iran.
Meanwhile a spokesman for the Foreign and Commonwealth Office confirmed foreign secretary Dominic Rabb had been tested for coronavirus this week after feeling unwell.
"He followed NHS advice, was tested for coronavirus and self-isolated. The test was negative and he was straight back to work," the spokesman added.
The World Health Organisation said the outbreak is "getting bigger" as more countries report their first cases.
It has now raised the risk level to its maximum of "very high", meaning an "immediate response" - within hours - is required as soon as a case is suspected.
However, new cases in China, where the virus began, are continuing to slow.