The new, October 31 cutoff date averts a precipitous and possibly calamitous Brexit that had been scheduled for Friday.
The extension at least provides more time for May to break the logjam, though it could easily just prolong the national agony.
However senior Tories warned the pressure on her to go would "increase dramatically" amid frustration among MPs that there is still no certainty as to when Britain will leave. "Gauke stated the prime minister had been" representing" on the potential for bringing it back to Parliament for a vote.
Mrs May spoke to the 27 European Union leaders for just over an hour, before they met for dinner without her to decide Britain's fate. Compared to some testy recent summits, there were indications of warmth and humor. May and German Chancellor Angela Merkel were filmed laughing above a tablet bearing a picture showing the two of them speaking on Wednesday while sporting blue jackets that were similar.
The summit was more tense than expected, with France's President Emmanuel Macron the strongest voice opposing a long extension as the talks stretched from early evening to early Thursday morning. He also said that this relative calm would make a People's Vote the most likely outcome.
Tusk said that during the extension Britain "will continue its honest cooperation as a full member state, with all its rights, and as a close friend and trusted ally in the future".
The European Parliament will need to accept any bargain May reaches the EU to ensure it is binding.
However, U.K. lawmakers rejected it three times.
The British Parliament has repeatedly rejected a withdrawal deal negotiated with the European Union, leading to the deadlock over Britain's long-awaited departure.
But the delay does not solve Britain's Brexit conundrum.
"She has previously said that" as prime minister" that she could not agree to let Britain stay in the European Union beyond June 30, and she's also promised to step down after Brexit is sent.
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However it is understood she still intends to stick to her promise and remain as PM until the next phase of the Brexit negotiations begin - whenever that may be.
Every initiative has floundered so far.
The delay avoids a possible economic calamity on both sides of the Channel but does little to resolve the political morass that has seen May's control over her MPs and cabinet gradually slip.
Labour, which seeks to retain close economic ties with the European Union after Brexit, accuses the government of failing to offer concrete changes to its Brexit blueprint.
Talks over the past week have not made a breakthrough, though they continued on Thursday, and neither side seemed willing to abandon them yet.
The battle starts in Britain over things to do with all the time. Opposition parties have been now baying for an early general election.
In the working-class port town of Tilbury east of London, which strongly backed leaving the European Union in the 2016 referendum, Brexit supporters voiced their frustration at the new delay.
Meanwhile the government says it will "continue to make all necessary preparations" for a no-deal Brexit.
Pro-Brexit lawmakers said it was time to replace May with another leader a Brexiteer.
"It was an amicable debate", Dutch prime minister Mark Rutte told reporters after the summit. Many of them find the prospect of Britain participating in the European elections unpalatable, given that it will be nearly three years since Britain voted to leave the EU.
"She feels trapped", explained Thomas Raines, head of the Europe schedule at the Chatham House think tank.