That plan was scuttled Monday by House of Commons Speaker John Bercow, who declared that Parliament can only vote again if the deal is "fundamentally different" to the version rejected by 230 votes in January and 149 votes last week.
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"If the Government wishes to bring forward a new proposition that is neither the same nor substantially the same as that disposed of by the House on March 12, this would be entirely in order", Bercow said.
Meanwhile, the EU's chief Brexit negotiator, Michel Barnier, hinted on Tuesday that Brussels will demand a clear answer to the question of how the government intends to proceed as the quid pro quo for granting an extension. "This clearly increases the chance of a no deal Brexit but from a legal perspective, the extension can be granted right up until the deadline".
But that's not going to be an easy task for Mrs May, with her cabinet divided over whether a long extension is acceptable, and parliament gridlocked after the Speaker warned that he might not allow a further vote on Brexit without substantial changes. "Will the United Kingdom request an extension because it wants a bit more time to rework the political declaration?"
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Bercow's decision came as May was negotiating with the Democratic Unionists (DUP), the small Northern Irish party which props up the Conservative Party in government, over the terms by which its MPs could back her deal. So there is a "backstop" which would kick in if Britain and the European Union fail to strike a deal.
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Any extension would have to be approved by a vote in both houses of parliament next week, because the 29 March date is written into the EU Withdrawal Act.
May's deal, agreed with Brussels, has been heavily rejected twice in the United Kingdom parliament, by 230 and 149 votes, respectively.
He declined to say how long a delay she would request, or for what objective, simply insisting, "you're going to have to wait for that letter to be published".
The Prime Minister's official spokesman said: "What you can see from the Prime Minister and her colleagues is an absolute determination to find a way in which Parliament could vote for the United Kingdom to leave the European Union with a deal".
House of Commons Speaker John Bercow rules out a third vote on the same Brexit deal, 18-04-2019. The very future of our Nation is at stake so the option of ignoring them is not available to us.
May had warned MPs that unless they approve the Brexit divorce deal, Britain's departure from the European Union could face a long delay - which upset some Conservative MPs fearing a softer Brexit.
"We need an initiative, we need something new because if it's an extension to remain in the same deadlock [that would be unacceptable.] How do we get out of this deadlock?"
The BBC's political editor, Laura Kuenssberg, said the prime minister's letter will also include a provision for a delay of up to two years.