The proposal approved by the Commons means she has the Herculean task of trying to convince European politicians to reopen the Brexit deal they spent 19 months negotiating, in order to rewrite the hated Irish backstop.
Within minutes of the result being announced the most senior European politicians completely ruled out any renegotiation, setting the stage for a major clash between the UK and the EU.
The result in parliament came on a night when the government also defeated a string of attempts to seize control of the Brexit process, though rebel MPs did manage to pass one amendment which saw the Commons take a symbolic stance against a no-deal exit.
In a further development, Jeremy Corbyn also finally agreed to meet Ms May to discuss ideas on how the country should move forward.
MPs approved the amendment tabled by the chair of the backbench 1922 Committee, Sir Graham Brady, by 317 votes to 301 to accept Ms May’s deal as long as an alternative to the backstop can be found.
The majority was one of the most narrow of the evening, and represented the devilishly complex situation facing the Commons on Brexit – with seven Labour MPs backing senior Conservative Sir Graham’s amendment, and eight Tories opposing it.
Ms May said she and other members of the government would now be speaking with the European Union about “how we address the house’s views” on Brexit.
After the result came in the prime minister told the Commons: “A fortnight ago this house clearly rejected the proposed withdrawal agreement and political declaration, with just 202 members voting in favour.
“Tonight a majority of honourable members have said they would support a deal with changes to the backstop.
“It is now clear that there is a route which can secure a substantial and sustainable majority in the house for leaving the EU with a deal.”
Earlier in the evening a plan put forward by Labour MP Yvette Cooper to prevent a no-deal Brexit in March by giving MPs the power to request an extension to Article 50 failed to receive the backing of the Commons, being defeated by 321 votes to 298.
An amendment tabled by former Tory attorney general Dominic Grieve that would have seen MPs seize control of the Brexit process, by enabling backbenchers as well as Ms May’s executive to table legislation between now and March 29, was also defeated by the government by 321 votes to 301, a majority of 20.
The pound dipped sharply after the failure of the attempts to delay Brexit, losing around 0.7 per cent against both the US dollar and euro, though there were indications later that it may be rising.
The only loss Ms May suffered was on a motion that said the UK would not leave the EU without a deal – by 318 votes to 310 – but the amendment tabled by former Tory minister Dame Caroline Spelman was only advisory and had no legal force.
Mr Corbyn said because the Spelman amendment had passed, he was now willing to meet Ms May, having previously ruled out the possibility.
Tory MP Nick Boles, who had teamed up with Ms Cooper on her amendment, said the pair would “consider what amendments will be needed” if Ms May fails to win any further concessions from Brussels.
Former foreign secretary Boris Johnson said that if the EU would not reopen the withdrawal agreement he would support the prime minister if she chose a “managed no-deal” Brexit.
He told Sky News: “I certainly think there are plenty of alternatives if they won’t give us the change we need.
“I think it is possible to go for a different type of agreement, a much more bare bones agreement – some would call it a managed no deal.”
Several Tory MPs told The Independent they expected to be back in the Commons to vote against Ms May’s deal, as they did not believe she would be able to secure sufficient changes.
One said: “I can’t think of a bigger understatement, than the PM saying there isn’t much appetite in Brussels to rewrite the backstop.”
The response from Brussels was immediate. European Council president Donald Tusk’s spokesman said “the withdrawal agreement is not open for renegotiation”, and the European parliament’s chief Brexit negotiator, Guy Verhofstadt, said there was “no majority to reopen or dilute” the deal.
French president Emmanuel Macron also said the agreement was “not renegotiable”, something echoed by the Irish government: “The best way to ensure an orderly withdrawal is to ratify this [current] agreement.”
Downing Street aides admitted the task ahead looked difficult, but highlighted that the previous deal had been heavily defeated and that the Commons had shown what needed to happen for the deal to pass.
If Ms May does secure any changes to the deal she will put it to another “meanignful vote”. If she does not then she must return to the Commons by 13 February to give MPs another chance to propose alternative measures.
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