Theresa May has been granted an extra two weeks to come up with a Brexit solution after talks with EU leaders.
The UK’s departure date had originally been set for 29 March.
If Mrs May can get her withdrawal deal through Parliament next week, that date will be pushed back to 22 May to give time to pass the necessary legislation.
If the prime minister can’t get the deal through, the UK will have to propose a way forward by 12 April for EU leaders to consider.
European Council President Donald Tusk said all Brexit options would remain open until then.
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“The UK government will still have a choice between a deal, no deal, a long extension or revoking Article 50,” he said.
“The 12 April is a key date in terms of the UK deciding whether to hold European Parliament elections.
“If it has not decided to do so by then, the option of long extension will automatically become impossible.”
Mrs May ruled out revoking Article 50, which would cancel Brexit, and she also said “it would be wrong” to ask Britons to vote for candidates for the elections to the European Parliament, due to be held from 23-26 May, three years after they voted to leave the EU.
The UK’s departure date is still written in to law as next Friday, 29 March.
29 March: Current Brexit date in UK law
12 April: If MPs do not approve the withdrawal deal next week – “all options will remain open” until this date. The UK must propose a way forward before this date for consideration by EU leaders.
22 May: If MPs do approve the deal next week, Brexit will be delayed until this date
23-26 May: European Parliamentary elections are held across member states
Mrs May is expected to table secondary legislation – that has to go through the Commons and the Lords by next Friday – to remove 29 March from UK law.
But Downing Street sources say an agreement with the EU to extend the Brexit deadline would be a piece of international law and would take precedence even if Parliament rejected it.
Mrs May said MPs had a “clear choice”.
Speaking on Thursday, after waiting for the 27 other EU countries to make their decision at a summit in Brussels, the prime minister said she would now be “working hard to build support for getting the deal through”.
MPs are expected to vote for a third time on the Brexit withdrawal deal next week, despite Commons Speaker John Bercow saying what is put forward must be substantially different to be voted on.
With only eight days to go until the scheduled Brexit day, what worried EU leaders most was Mrs May’s inability – or refusal – to answer their insistent question: what will you do if the Brexit deal fails to get through Parliament next week?
It was then that EU leaders decided they had to take control of the situation if they hoped to head off a no-deal Brexit.
By the time the 27 leaders emerged at their Brussels summit, bleary-eyed from hours of bad-tempered debate about delaying Brexit, it became clear that they had “done a May”.
By that, I mean they had managed to kick the Brexit crunch-time can another couple of weeks down the road – something Theresa May has become famous for throughout the Brexit process.
At a news conference on Thursday night, Mrs May also referred to her speech from Downing Street the previous evening, which had sparked an angry reaction from MPs after she blamed them for the Brexit deadlock.
“Last night I expressed my frustration and I know that MPs are frustrated too,” she said. “They have difficult jobs to do.
“I hope that we can all agree we are now at the moment of decision. And I will make every effort to ensure that we are able to leave with a deal and move our country forward.”
BBC political editor Laura Kuenssberg said Mrs May, although she did not apologise, had shown a “very different tone to MPs”.
But she added that the PM was not drawn on what she would do if her deal fails again in a vote next week.
Mrs May arrived at the EU summit in Brussels on Thursday hoping to persuade the EU to postpone Brexit from next Friday – the date which is set in law – to 30 June.
She made her case in a 90-minute presentation to her European counterparts.
BBC Europe reporter Gavin Lee said that one EU source inside the room told him that some leaders felt she was “obfuscating”.
Several leaders said they felt “surprised” that Mrs May appeared to be “seriously contemplating a no-deal scenario”, the source added.
Mrs May then left the room, and discussions between EU leaders ran late into the evening.
In her briefing to journalists, Mrs May dismissed calls to revoke Article 50 – the process by which the UK leaves the EU – which would mean Brexit is cancelled. It comes after a petition calling for Article 50 to be revoked passed two million signatures. A march demanding another referendum is also planned for Saturday in central London.
Mrs May said people had voted to leave and were told their decision would be respected.
Brexit: Departure date pushed back by at least two weeks}