The UK has seen its biggest day-on-day increase in coronavirus cases, bringing the total number to 87.
Almost all of the 36 new patients had recently travelled to affected countries or been infected by others who had done so, the UK's chief medical adviser Prof Chris Whitty said.
But it is not clear how three new patients in England caught the virus.
Meanwhile, it has emerged that two patients at King's College Hospital have been diagnosed with the virus.
A family member of an elderly patient at the hospital told the BBC one of those who was infected had been staying in the same room as his relative prior to being diagnosed.
The hospital said it had restricted access for staff and visitors to the affected ward, and was tracing those who may have come into contact with the infected patients.
Of the 87 cases in the UK there are 80 in England, three in Scotland, one in Wales and three in Northern Ireland.
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Two of the latest patients are Scottish, with one living in the Grampian area and the other in Ayrshire. One had recently travelled to northern Italy while the other had contact with a known positive case.
New cases of the virus in England have been confirmed in Liverpool, York, Carlisle, Newcastle, Torquay and Manchester, as well as in Lancashire and Derbyshire.
In Northern Ireland, a Queen's University postgraduate student, who recently returned from northern Italy and has been mixing with fellow students, tested positive.
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Meanwhile, students at Goldsmiths in south London have been told a visitor to one of the university's halls of residences has coronavirus.
In an email, the university said the person was "being looked after" and that the student they visited was "self-isolating as a precautionary measure".
Buckingham University and the University of London have put back their graduation ceremonies to minimise the spread of the disease.
Prof Whitty – who earlier said an epidemic in the UK was "likely" – said people who had been in contact with the new patients are being traced.
In "the worst case scenario", up to 80% of the UK population could be infected with coronavirus, which causes the disease Covid-19, he said.
But he said a move to lock down entire towns and cities across the UK would be "very unlikely".
In other key developments:
- Health Secretary Matt Hancock chaired a meeting of the emergency Cobra committee in London
- The release of the new James Bond film has been put back by seven months
- Hundreds of thousands of businesses may not be able to claim on losses incurred by the coronavirus outbreak, despite a government U-turn
- Hospitals in England have been asked to carry out more consultations by video to reduce the risk of the virus spreading
- Italy is closing schools and universities across the country until mid-March. Italy is the worst-hit European country with some 2,706 cases and 107 deaths so far
- Guests who left a quarantined hospital in Tenerife have been told to self-isolate for a further week after a British woman at the hotel tested positive on 2 March
- CCTV images have been released of four men wanted over an alleged racist attack on London's Oxford Street, reportedly motivated by virus fears
- There are fears of global shortages of some common drugs after India limited the export of certain medicines
- Grants and low-cost loans are part of a $12bn (£9.4bn) emergency aid package to help developing countries
- And Australians have been stockpiling toilet paper after the country's first death was reported
A major public health campaign urging people to wash their hands regularly for at least 20 seconds has also been launched.
Health Secretary Matt Hancock said washing hands regularly was the "single most important thing that an individual can do".
Public Health England says to use a hand sanitiser with at least 60% alcohol if soap and water are not available, advising that hand washing is particularly important after using public transport.
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