Theresa May has blamed MPs for failing to implement the result of the 2016 EU referendum, and told voters who want Brexit to be over: "I am on your side."
In a televised address from Downing Street, Mrs May said it was "a matter of great personal regret" for her to have to ask for a three-month delay to Britain's withdrawal from the EU, which was due to take place on March 29.
She will go to Brussels today to make a formal request to the other 27 EU leaders for an extension to the two-year Article 50 negotiation process.
Earlier, European Council president Donald Tusk said he believed a short delay "would be possible" after he spoke to the prime minister by phone.
However, he said the extension - which must be agreed unanimously by the EU27 - was likely to be conditional on Mrs May succeeding in forcing her twice-rejected Brexit deal through parliament.
She made the request in a letter to Mr Tusk exactly 1,000 days after the 2016 referendum which delivered a 52pc-48pc majority to quit the EU.
Speaking behind a lectern in 10 Downing Street, Mrs May said MPs - who rejected her deal by 230 votes in January and 149 last week - had been "unable to agree on a way to implement the UK's withdrawal".
In a message directed at voters, she added: "Of this, I am absolutely sure: You the public have had enough.
"You are tired of the in-fighting, you're tired of the political games and the arcane procedural rows, tired of MPs talking about nothing else but Brexit when you have real concerns about our children's schools, our National Health Service, knife crime.
"You want this stage of the Brexit process to be over and done with. I am on your side. It is now time for MPs to decide."
Mrs May said MPs had so far done "everything possible" to avoid making a decision on the way forward.
She warned that an extended delay means staging "bitter and divisive" elections to the European Parliament at a time the country needed bringing back together.
"I passionately hope MPs will find a way to back the deal I have negotiated with the EU.
"I will continue to work night and day to secure the support of my colleagues, the DUP and others for this deal, but I am not prepared to delay Brexit any further than June 30."
In his statement, Mr Tusk said it should be possible for EU leaders meeting in Brussels to approve an extension to Article 50, though the "question remains open" as to the duration.
He said it should then be possible to finalise the deal without the need for another summit - provided it secured the backing of MPs.
After the tumultuous events of recent weeks, he appeared to acknowledge the difficulties facing the prime minister, describing the hopes of success as "frail, even illusory".
"Although Brexit fatigue is increasingly visible and justified, we cannot give up seeking until the very last moment a positive solution," he said.
"We have reacted with patience and goodwill and I am confident that also now we will not lack the same patience and goodwill at this critical point."