The UK is "coping very well indeed under the most challenging possible circumstances", the Prime Minister has said.
Boris Johnson said the Government was "massively ramping up" testing for coronavirus as he announced that 405,000 people have now signed up as volunteers to help vulnerable people.
He said it was hoped that "very soon" 250,000 tests would be carried out each day.
Speaking at the daily press conference inside Number 10, he added that he wanted to offer a "special thank you to everyone who has now volunteered to help the NHS".
He added: "When we launched the appeal last night, we hoped to get 250,000 volunteers over a few days.
"But I can tell you that in just 24 hours, 405,000 people have responded to the call.
"That is already, in one day, as many volunteers as the population of Coventry."
The mass sign-up follows Health Secretary Matt Hancock's request for a quarter-of-a-million people to donate their time to help the 1.5m people isolating for 12 weeks in an attempt to slow the spread of Covid-19.
NHS chief executive Sir Simon Stevens praised the "overwhelming response" to the call for volunteers to help the vulnerable and thanked those who will devote their time to the health service.
The figure was revealed as the Foreign Office announced that Steven Dick (37), the deputy head of mission at the British Embassy in Budapest, had died after contracting coronavirus.
During the press conference, England's chief medical officer Professor Chris Whitty dismissed claims made earlier by Professor Sharon Peacock, director of the National Infection Service at Public Health England (PHE), who said members of the public will be able to carry out coronavirus antibody tests at home very soon.
She said 3.5m tests - which will tell people whether they have had the virus - would be made available and distributed via Amazon and in places like Boots.
But Prof Whitty said members of the public would not be buying these tests via the internet next week. He stressed that frontline NHS workers would need the tests first so they could get back to work if they have already had the virus.
The NHS is also carrying out separate tests to see if people currently have the virus, which are being given to patients in hospitals alongside some community sampling.
Earlier, NHS England's medical director, Professor Stephen Powis, said hundreds of thousands of these tests for Covid-19 per day could become a reality within weeks.
It comes as:
◊ Boris Johnson urged London Mayor Sadiq Khan to "get more Tubes on the line" but resisted calls to ban non-essential construction workers from heading to building sites.
◊ Parliament is set to adjourn for an early Easter break after emergency legislation to tackle Covid-19 is approved.
◊ The Prime Minister confirmed that ministers are considering asking black taxi drivers to act as a transport service for NHS workers.
◊ A total of 435 patients who tested positive for coronavirus had died in the UK as of yesterday.
Prof Whitty said there were shortages along many supply chains in the production of tests because "every country in the world is simultaneously wanting this new thing".
He added: "It's not that there is no testing going on, what we need, clearly, is to be able to scale it up."
Replying to a question about how well the country was coping, Mr Johnson said that "never in our history has the Government put its arms around people in the way we are doing now to help them get through this time".
He added that a tailored package of support would be announced today to help self-employed people.
The PM said: "I do think when you look at the sheer scale of what the Government is doing to get this country through, we will cope and are coping very well indeed under the most challenging possible circumstances.
"To come out of it well together as I know we can, we all need to follow the instructions the Government have given and to stay at home, protect the NHS and that's the way to save lives."