Mr Gove said Boris Johnson, who has mild symptoms and has tested positive for the illness, will continue to lead the fight against the spread of Covid-19.
He announced a "new alliance" between businesses, researchers, and universities that will see hundreds of frontline workers tested for the disease by this weekend.
Critical care nurses, emergency department medics, paramedics, and GPs will be first in line, with the number of tests being done set to double by this time next week.
It is antigen testing rather than antibody testing, so will tell medics if they have the disease, but not if they had recovered from it and are therefore thought to be immune - and safe to go back to work.
Two more field hospitals will also be built in Birmingham and Manchester, the briefing was told by the NHS's chief executive Sir Simon Stevens, with more to follow.
That's in addition to one being built in London, which will have room for 4,000 coronavirus patients.
Sir Simon said hospitals across the UK had been "reconfigured" and now had room for 33,000 Covid-19 patients.
The jump in total coronavirus-related deaths in the UK from 578 to 759 today was an increase of 181 - the biggest day-on-day rise in deaths since the outbreak began.
It represents an increase of 31 per cent on the total yesterday.
It took 13 days for the number of deaths in the UK to go from one to just above 100. It took a further eight days to reach the latest total of 759.
The number of people tested in the UK for coronavirus has seen its biggest day-on-day increase since the outbreak began: up 8,911, from 104,866 tested as of 9am on March 26 to 113,777 tested as of 9am on March 27.
Just under 50,000 tests (46,801) were carried out in the seven days to 9am March 27.
In the previous seven days the number was 34,205.
The daily number of new confirmed cases of coronavirus has reached a new high, with 2,921 recorded in the 24 hours to 9am March 27.
The equivalent number for the previous 24 hours was 2,129.
Mr Gove said that Mr Johnson and Mr Hancock both testing positive for coronavirus shows that the "virus does not discriminate", adding: "We are all at risk."
Prof Whitty said he was self-isolating after experiencing symptoms, but did not say if he had been tested.
Mr Gove said: "Today I can announce that the Prime Minister has brought together businesses, research institutes and universities in a new alliance to boost testing capacity for frontline workers.
"This will be antigen testing, testing whether people currently have the disease, so that our health and social care workers can have security in the knowledge that they are safe to return to work if their test is negative.
"These tests will be trialled for people on the frontline starting immediately, with hundreds to take place by the end of the weekend, dramatically scaling up next week."
Mr Gove thanked those working in the NHS and "all those involved in this effort to reinforce the frontline in the battle against the virus".
And he said: "The best scientific analysis now is that the rate of infection has been doubling every three to four days."
Sir Simon thanked the public for the Clap For Carers effort, saying it would have meant an "enormous amount" to NHS staff to know the "whole country is behind them".
He said in the last two weeks 18,000 doctors and nurses have rejoined the health service after "answering the calls to arms".
"It was therefore very gratifying for our staff across the NHS to see this remarkable outpouring in the Clap For Carers last night," he said.
"For many nurses coming home from a day at hospital or for other staff returning to start again for the night shift it would have meant an enormous amount to know that the whole country is behind them."
Currently, there are 6,200 confirmed Covid-19 patients being treated in English hospitals, and Sir Simon said that number is "only bound to rise in the coming days".
He said testing of frontline NHS staff to determine whether they have or have had coronavirus will start next week.
He added: "From an NHS perspective, we think it is urgently important that we are able to test frontline workers who are off sick or otherwise isolating.
"That's why the work that Public Health England has been leading is so important because it means we are going to be able to double this time next week the number of tests we have been doing this week.
"I can say that today we will be rolling out staff testing across the NHS, starting next week with the critical care nurses, other staff in intensive care, emergency departments, ambulance services, GPs.
"As testing volumes continue to increase, we want to widen that to essential public service workers, as well as our social care workers, and continue with patient testing that is so vital."
Mr Gove said: "The fact that the virus is no respecter of individuals, whoever they are, is one of the reasons why we do need to have strict social distancing measures so that we can reduce the rate of infection and reduce the pressure on the NHS."
Sir Simon said that as of Thursday, there were just under 3,000 empty and available hospital beds in London, and there would be additional beds next week at the new NHS Nightingale Hospital.