Thousands of people are still in hospital with serious coronavirus symptoms - and hundreds are being admitted to hospitals every day.
The figures were published by Health Secretary Matt Hancock on the day that new lockdown rules came into effect.
Speaking at Monday's Downing Street press conference, he said: "The disease is not done yet".
New rules that came into force today allowed people to meet outdoors in groups of up to six, including in private gardens.
Many schools admitted more pupils and outdoor shops such as markets were allowed to open.
But official figures showed:
- 7,541 people are still in hospital with coronavirus
- 606 of them are on mechancial ventilators to help their breathing
- 479 people were admitted to hospital on May 30
- 111 deaths were confirmed on June 1, bringing the total death toll to 39,045
- And 1,570 new cases were confirmed on June 1
Mr Hancock also said the Government had the capacity to test far more people. He said: "Tests are available right now on the website.
"It's so important that people come forward for a test, and anyone who needs a test can get a test."
Mr Hancock said the "vast majority" of people found to have coronavirus were now being contacted by staff from the Government's new test and trace service. They are asked who they have been in contact with in recent days, so that those people can be asked to go into isolation.
Mr Hancock also said lockdown measures could be reimposed nationally if necessary.
He said: “We are attempting to move the system from these national, blanket measures to a more targeted approach – this is why test and trace is such an important part of that.
“But we have always said that we are prepared to reintroduce measures – whether that is nationally or in response to a localised outbreak – if that is necessary.”
Mr Hancock said there was a range of measures available to combat local flare-ups of coronavirus.
It could mean “shutting to new admissions a hospital A&E if there was an outbreak in that hospital”, he said.
The powers available were as broad as the “legal toolkit” that was used for the national lockdown, he said.
Local directors of public health would work with regional Public Health England and NHS teams “to make sure we got the response right”.
The Joint Biosecurity Centre would have a national role “to provide the advice and the information that would then be acted on locally”.
It would advise the UK’s chief medical officers who would report to ministers and local health bodies.
Earlier, Business Secretary Alok Sharma insisted the lockdown is being eased in a “very cautious” way as thousands of children in England began returning to school despite concerns from public health officials and parents.
He said he understands parents’ concerns over sending their children back to class on Monday, but added that the Government had not undertaken a “dash” to re-start the economy.
Mr Sharma told BBC Breakfast: “This is not a dash. These are very cautious steps that we are taking. They are phased.”
He said that he “completely” understands that “every parent wants to keep their child safe”, but insisted the Government had taken steps to ensure schools are safe to return to as classes re-opened to children in Reception, Year 1 and Year 6 in England.
However, the Association of Directors of Public Health (ADPH) warned that experts were “increasingly concerned” that ministers are making the wrong judgment by easing restrictions too quickly.
And the general secretary of the Association of School and College Leaders, Geoff Barton, said head teachers were reporting attendance rates of between 40-60% of eligible pupils, as parents kept children at home.
A survey of councils by the PA news agency has found dozens of local authorities across England, predominantly in the north, were advising against a return to school on Monday amid safety concerns.
Some have raised concerns that the test and trace programme is not yet “robust enough” to sufficiently reduce Covid-19 transmission in schools, where social distancing is hard to maintain with children.
But in some areas of England – such as Kingston and Richmond in London – local authorities report the vast majority of primary schools are providing some provision for priority year groups from Monday.
Schools, colleges and nurseries closed 10 weeks ago due to the Covid-19 outbreak, remaining open only for vulnerable youngsters and the children of key workers.
Coronavirus has infected more than six million people across the world and killed more than 370,000, according to a tally kept by Johns Hopkins University.
