Speaking in Downing Street, Mr Johnson told reporters: “I believe that a combination of the measures that we’re asking the public to take and better testing, scientific progress, will enable us to get on top of it within the next 12 weeks and turn the tide.
“I cannot stand here and tell you that by the end of June that we will be on the downward slope.
“It’s possible but I simply can’t say that that’s for certain,” he added.
“We don’t know how long this thing will go on for. But what I can say is that this is going to be finite.”
Earlier Sir Patrick Vallance, the UK’s chief scientific officer, was also asked how long it would go on for.
“We are dealing with a brand new virus. We are dealing with learning as we go on and I think now to put absolute timelines on things is not possible.”
In the press conference on Thursday evening, Mr Johnson ruled out closing down public transport in London but pointed out people in some parts of London were not following government guidance on social distancing and would be “enforced” to do so if necessary.
“I know how difficult it may be or it may seem right now, but if we do this together, we will save many, many thousands of lives,” he added.
He also urged businesses to stand by their employees “because we will stand by you”, adding that his chancellor would be making further announcements on Friday.
He also said the government has bought a test which can detect whether someone has had corona virus – and their immunity to it.
The press conference came as Italy’s death toll rose by 427 to 3,405, overtaking China’s toll.
In the UK, 144 people with corona virus have died, and 3,269 people have tested positive for it.
Speaking alongside the PM, Prof Chris Whitty, the UK’s chief medical adviser, warned there would be a “lag” before the public’s efforts to stem the spread of the virus would slow down case numbers.
He said there would be a “global issue” in the supply of personal protective equipment (PPE), saying NHS workers were right to complain about the shortages.
The Queen, who is now in Windsor Castle with the Duke of Edinburgh, said in her statement that the world was “entering a period of great uncertainty” and every individual had “a vitally important part to play”.
“Many of us will need to find new ways of staying in touch with each other and making sure that loved ones are safe – I am certain we are up to that challenge,” she added.
“You can be assured that my family and I stand ready to play our part.”
Even if the tide starts to turn in 12 weeks and the number of cases starts to fall, then we will still be far from the end.
It can take a long time for the tide to go out.
It is clear the current strategy of shutting down large parts of society is not sustainable in the long-term.
But the corona virus is not going to disappear.
For as long as large numbers of people in the UK have no immunity then cases will soar as soon as restrictions are lifted.
A vaccine would help, but that could be 18 months away.
If enough people are exposed and become immune then that would help too, however, this is likely to take even longer than a vaccine.
Efforts to rigorously test and then isolate anyone infected can help suppress the numbers of people infected. But this is what we were doing just a few weeks ago.
We may enter a period of restrictions being lifted and then re-imposed until a long-term solution is found.
From Friday, all schools in the UK will close their doors to nearly all pupils, except vulnerable ones and the children of key workers.
A full list of key workers, likely to include NHS workers, school staff and delivery drivers, is expected later.