The EU's spurning of May in Brussels is likely to worsen her plight in London, where she's being described as a "zombie leader" and is still struggling to get a hold on her party, despite overcoming a challenge Wednesday to unseat her as Conservative party leader - and therefore prime minister.
Theresa May is heading to Brussels for the second time this week in another frantic bid to win Brexit concessions after surviving a vote of no confidence by Conservative MPs.
The Conservative leader this week postponed a parliamentary vote on the deal in the face of huge opposition among MPs, including in her own party, who promptly launched a confidence vote against her.
The secret ballot was triggered by 48 of her MPs angry at her Brexit policy, which they say betrays the 2016 referendum result.
In her end-of-summit press conference, she acknowledged that the Withdrawal Agreement could not be reopened, but insisted their discussions had shown there was scope for "further clarification" on the way the backstop would operate.
The Brexit withdrawal agreement only talks about "best endeavours" being used to reach an agreement during the transition period.
Lib Dem leader Sir Vince Cable, who is against Brexit, told BBC Breakfast: "We are still back with the problem that the government has a proposal that we can't get through Parliament and we have got to try and break that gridlock".
European Commission president Jean-Claude Juncker told reporters afterwards: "Our UK friends need to say what they want instead of asking us what we want".
She achieved high office (Home Secretary) thanks to David Cameron, who failed to gain a majority in Parliament at the 2010 election, in spite of the fact that Labour had been in power since 1997 and that the outgoing prime minister, Gordon Brown, was deeply unpopular.
"If the British prime minister thinks one or another additional explanation can be helpful before she brings it to a vote, then we should do that".
This is because the deal that she has negotiated with the European Union creates precisely the conditions which guarantee that her objective of a free trade deal with the European Union, first stated in January 2017 and repeated on many subsequent occasions, can not be achieved.
Rees-Mogg and other eurosceptics hate the divorce deal May agreed with the European Union last month, which they fear risks tying Britain to the bloc for years after Brexit on March 29.
While May's position as party leader - and prime minister, as long as there is no snap election - is safe for another year, her struggle is far from over.
Labour leader Jeremy Corbyn said: "The last 24 hours have confirmed that Theresa May's Brexit deal is dead in the water". These people have to vote with the government or they lose their government job and any chance of future preferment.
Speaking after Prime Minister's Questions, a Downing Street source said May would tell MPs to focus on the damage that would be caused to the Brexit negotiations if there was a change in Prime Minister.
"I think it is necessary that something new is put on the table".