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4 year oldMr Johnson, 55, has been given oxygen but has not been put on a ventilator, cabinet minister Michael Gove said.
Foreign Secretary Dominic Raab has been asked to deputise for the PM. Arriving at No 10 on Tuesday, he said the prime minister was "in very good hands".
World leaders have sent messages to Mr Johnson wishing him well.
The prime minister, 55, was admitted to St Thomas' Hospital with "persistent symptoms" on Sunday evening and moved to the intensive care ward on Monday at 19:00 BST.
He was moved as a precaution so he could be close to a ventilator - which takes over the body's breathing process - BBC political correspondent Chris Mason said.
Mr Gove told BBC Radio 4's Today programme on Tuesday: "The prime minister's not on a ventilator. He has received oxygen support."
If there is any change in his condition "No 10 will ensure the country is updated", Mr Gove added.
The Queen has been kept informed about Mr Johnson's health, Buckingham Palace said.
As the first secretary of state, Mr Raab is the minister designated to stand in for Mr Johnson if he is unwell and unable to work.
Mr Raab arrived at No 10 on Tuesday morning and will later chair the government's daily Covid-19 meeting.
He said earlier there was an "incredibly strong team spirit" behind the prime minister and that he and his colleagues were making sure they implemented plans Mr Johnson had instructed them to deliver "as soon as possible".
Former Conservative leader Sir Iain Duncan Smith said the government will "continue to work" as decisions are made collectively by the cabinet.
He also sounded a warning to people who have broken social distancing guidelines, saying "if the most powerful man in Britain can come down with this, so can you".
Mr Johnson was initially taken to hospital for tests after announcing 11 days agothat he had the coronavirus. His symptoms included a high temperature and a cough.
Earlier on Monday, he tweeted he was in "good spirits".
Last night, on the advice of my doctor, I went into hospital for some routine tests as I’m still experiencing coronavirus symptoms. I’m in good spirits and keeping in touch with my team, as we work together to fight this virus and keep everyone safe.
— Boris Johnson #StayHomeSaveLives (@BorisJohnson) April 6, 2020
Mr Johnson's friend and former direction of communications Will Walden told BBC Radio 4's Today programme Mr Johnson is "far fitter than he looks".
"He will whip anybody's backside on a tennis court, he runs regularly, he doesn't smoke, he drinks moderately.
"So I think if anyone is in a good position both physically and mentally to fight off the disease then the prime minister is that person."
After very, very little information was shared today, the prime minister was taken into intensive care at around 19:00 BST.
We've been told he is still conscious, but his condition has worsened over the course of the afternoon.
And he has been moved to intensive care as a precaution in case he needs ventilation to get through this illness.
The statement from Downing Street makes clear he is receiving excellent care and he wants to thank all of the NHS staff.
But something important has changed, and he has felt it necessary to ask his foreign secretary to deputise for him where needs be.
That is a completely different message from what we have heard over the past 18 hours or so, where it was continually "the prime minister is in touch" and "he is in charge" - almost like everything is business as usual.
But clearly being in intensive care changes everything.
It comes as the number of coronavirus hospital deaths in the UK reached 5,373 - an increase of 439 in a day.
The Department of Health and Social Care said there were now 51,608 confirmed coronavirus cases.
Also in hospital with coronavirus is veteran Labour MP for Rochdale Tony Lloyd, 70, who is being treated at Manchester Royal Infirmary.
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