May had promised the British people that Brexit would help them take back control from Europe. Instead, she found that her 27 fellow E.U. leaders took back decision-making from her, dictating a political calendar for the coming weeks that gives both sides time to prepare for the worst-case scenario.

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If British lawmakers approve a divorce deal they have twice rejected, they can leave the European Union on May 22. If they reject it, they must plan an alternative by April 12 or fall off the same no-deal cliff that evening.

Either way, the decisions will be forced earlier than the three-month delay May was seeking in Brussels.

May told reporters that she was satisfied with the extension. She said it was now “very clear” what the options and deadlines are for British lawmakers.

“What the decision today underlines is the importance of the House of Commons passing a ­Brexit deal next week so that we can bring an end to the uncertainty and leave in a smooth and orderly manner,” she said.

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May appeared relatively upbeat and matter-of-fact, and took pains to reach out to lawmakers. Gone was the hectoring, accusatory attitude of the night before when she lashed out at Parliament and said it was lawmakers’ fault that Brexit was being delayed.

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But European leaders made little secret that they were fed up.

“The E.U., to put it clearly, is facing a British political crisis,” said French President Emmanuel Macron, who argued for an uncompromising stance during the talks. “British politicians are unable to deliver what the people asked them to do. The people voted for Brexit, but we are in a situation where the British Parliament is telling us they are not voting for the agreement we negotiated for two years.”

Leaders felt May was so removed from political realities that one European diplomat said her assessments were “a bit like coming from another planet.” The official and others spoke on the condition of anonymity to recount the talks.

In footage of the E.U. leaders greeting one another in Brussels before settling in for business, May joked with Luxembourg’s prime minister, who just minutes before had threatened to cast Britain out of the E.U. without a deal. She exchanged a tense and unsmiling double-cheek-peck with European Commission President Jean-Claude Juncker, with whom she has tangled and who has taken a particularly hard line against her.

One leader after another went around a ring-shaped table to hammer May about her political strategy, her preparations for a no-deal departure and why she wanted three months more. On each question, they found her wanting, diplomats said, after she ate up much of her opening discussion time to read off a letter that leaders had already seen the day before. In all, she was grilled for nearly two hours — an eternity in the tightly choreographed world of high-stakes summits.

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“You could feel the patience running thin,” an official said.

After she was dismissed from the room, leaders talked and talked, disinviting May from a dinner of duck à l’orange to keep hammering out a deal. The Bulgarian ambassador to the European Union tweeted a photo of a scrum of senior officials clustered in a corridor, some of them crouching on the ground, as they fought back and forth over the details.

By late evening — after May had stewed outside the room for four hours — European Council President Donald Tusk delivered the news. She had won a reprieve — but not the one she requested.

The extension provides “so many options,” Tusk told reporters afterward. He said Britain will still have the choice of a deal, no deal, a long delay or canceling Brexit.

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Until April 12, he said, “all options will remain open and the cliff-edge date will be delayed.”

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The April date is the deadline to organize late-May elections for the European Parliament, a necessary step, E.U. leaders believe, if Britain is to stay inside the union beyond them.

The stuttering sequence of deadlines increases pressure on British lawmakers to cut across party lines to approve some sort of deal. It also gives E.U. leaders political cover with their own voters if Britain departs without a safety net, since they can say they did everything possible to avoid it.

Economists and political analysts have warned that a no-deal Brexit could result in the halting of trade and travel and a hit to the British and European economies.

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By now, many E.U. policymakers no longer hold out hope for a second referendum that could reverse the Brexit decision and think it best to break up and move on — a marked shift on a continent that once wanted Britain to stay.

“We don’t want in the coming months, in the coming years, to be busy with Brexit,” said Guy Verhofstadt, the European Parliament’s Brexit coordinator, who used to post wistful videos on Twitter appealing to Britons to abandon Brexit. “We want to be busy with the renewal of the European Union,” he told reporters before the meeting.

When May triggered the two-year Brexit window, confident ­Brexiteers said squabbling European countries would quickly succumb to the crack British team of negotiators. Instead, the Europeans have remained unusually united, while the first two British Brexit ministers resigned to protest the deals they themselves agreed on.

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May and British lawmakers are now blaming each other for failing to have agreed on how to leave.

May’s deal was defeated by 149 votes on March 12 — and by 230 votes on Jan. 15. Both votes were widely described as humiliations. Many members of her Conservative Party voted against the prime minister’s exit plan and have vowed to do so again.

May needs to persuade at least 75 lawmakers to back the exit deal, including the 10 members from Northern Ireland’s Democratic Unionist Party, which props up her minority government. She also needs at least some of the most rebellious, hardcore Tory ­Brexiteers, plus some Labour Party lawmakers, to cross over. This will not be easy.

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Some of the 48 percent of people who voted to remain in the European Union in the June 2016 Brexit referendum were growing increasingly nervous about what might happen over the next few days.

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An online public petition calling on May to cancel Brexit attracted more than 2 million signatures in mere hours, crashing the website once from the surge in traffic.