If the Government refuses to do so, Remain MPs could still try to force through a change in the law that would force ministers to delay Brexit to avoid no deal.
Tory MP Oliver Letwin and Labour's Yvette Cooper, who have tabled an amendment on Wednesday in order to prevent a no-deal, confirmed that even though May had conceded to "most" of the intentions of their amendment, they still meant to put it to a vote in order to avoid a June cliff edge.
Theresa May's dramatic announcement on Tuesday that she would allow MPs to vote on delaying the UK's European Union withdrawal beyond March 29 took the sting out of an evening of Brexit votes which had been expected to feature a number of ministerial resignations.
A symbolic amendment putting the plans to paper was expected to be waved through the Commons on Wednesday without a vote after the government said it would back it. Labour MP Yvette Cooper - who has led efforts to block no deal - claimed victory when David Lidington said the Government would be bound by a vote against no deal on March 13.
Any more significant Conservative bust-ups have now been delayed for up to two weeks, as Mrs May prepares to bring her Withdrawal Agreement back to the Commons for a "meaningful vote" by March 12, although there have been hints the showdown could be as soon as next week.
Under Labour's plan, the United Kingdom would remain in a customs union with the EU after Brexit and would aim for a close relationship with the single market.
British PM May offers to delay Brexit vote
If she doesn't find a way to remove the threat of a no-deal departure next month, May is likely to suffer a spate of resignations. May put off a vote in parliament on her deal to March 12 , just 17 days before Britain is due to leave the EU.
Alison McGovern MP, a leading supporter of People's Vote, said: "Once again, the government has had to scramble to avoid defeat on Brexit and today has shown that, when it comes to Brexit, parliament is able to agree what it is against - but in our system, it is hard for those other than the Government to say what it supports".
The Prime Minster's dramatic announcement on Tuesday that she would allow MPs to vote on delaying the UK's European Union withdrawal beyond March 29 took the sting out of an evening of Brexit votes which had been expected to feature a number of ministerial resignations. If this is rejected, there will be a vote to approve delaying Brexit on March 14.
May's climbdown took much of the heat out of a series of votes on Wednesday that could have ripped control of the entire process away from the government.
Mrs May's U-turn threw a spotlight on the Labour leader, whose "constructive ambiguity" on Brexit has long frustrated those in his party who back a so-called People's Vote, and played a part in the defection of eight MPs to the new Independent Group last week.
On the weekend the prime minister said MPs would have the opportunity to vote on it again by 12 March, a little more than two weeks before the United Kingdom is due to leave the European Union.
Speaking after he addressed his group of MSPs, he said: "If we are not successful in amending the Theresa May deal then we feel we are left with no option than to back a public vote.
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There will now be an extension of the Brexit deadline to allow for serious cross-party dialogue on a better deal to protect the British national interest".
There are 12 amendment by MPs on a raft of proposals.
The amendments included moves to rule out a no-deal scenario and backing of a second referendum.
MPs are staging a new round of non-binding votes on Brexit tonight - the third set since the deal was defeated on January 15.
"I don't support a second referendum, many of my Labour colleagues in the Parliamentary Labour Party - but also members as well as Labour leave voters - want the Labour Party to stand by its promise, and the promise was to respect the referendum", she said.
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