The government was forced into a major compromise on Tuesday when it had to agree to give parliament a greater role in Brexit negotiations to avert a defeat at the hands of rebels from her own party who want to keep close European Union ties after Britain leaves the bloc in March next year.
Some of the pro-EU rebels said they had been assured by May that elements proposed by Conservative lawmaker Dominic Grieve would be brought into the European Union withdrawal bill, which will sever ties with the bloc and "copy and paste" its laws.
But just an hour before the vote, due at around 1500 GMT, the government sought to compromise with senior pro-EU Conservative lawmaker Dominic Grieve, who had put forward his own amendment, which increased the risk of a government defeat.
Further votes on EU Withdrawal Bill amendments will take place on Wednesday, with no defeats expected for the government after ministers agreed a compromise wording over post-Brexit plans for a "customs arrangement".
"Let's remember what the Withdrawal Bill is for: it's about delivering a statute book that is ready for Brexit day", she said.
The amendment was one of 15 that peers had made to the bill as it made a bruising passage through the upper chamber.
"The government can not demonstrate the flexibility necessary for a successful deal if its hands are tied midway through that process", Davis said.
Part A said that if parliament rejects the final Brexit deal, the government would have to set out a new approach within seven days. They stood down after the government promised to engage in talks on a compromise.
This amendment gives Parliament control in the final stages of Brexit, preventing the government from signing a deal with the European Union that hasn't been approved by lawmakers. Rebels have said they will challenge May's plans to leave the customs union during votes on other bills, on trade and customs, which will be brought back to the house some time before July 24. Pro-Brexit members of the government want to be able to play the "no deal" card, but the House of Commons, where pro-EU voices are stronger, would nearly certainly reject the idea.
It was not as simple on Tuesday, when May was forced to defuse another rebellion in parliament by offering a compromise that could hand lawmakers more control over Brexit.
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"And she has given us that, and those discussions on how we are going to build on the amendment that has been approved by the House of Commons will start today".
A paper laying out the U.K. government position, due to be published this month, has been delayed because the Cabinet can not agree on a united stance.
The amendment on the customs union probably has majority support in the House of Commons, but the clause is so loosely worded that the government doesn't consider it binding.
The Daily Express featured the British flag as its front page with the headline: "Ignore the will of the people at your peril".
The Labour whips - who enforce discipline in the party - instructed lawmakers Wednesday to abstain on an amendment favoring membership of the EEA.
Davis said Parliament would be involved in the process but MPs could not undue the 2016 referendum.
"We are walking on a tightrope at the moment, we campaigned for remain but many of our MPs, including myself, now represent seats which voted heavily leave", John McDonnell, the party's economy spokesman and Corbyn's right-hand man said at an event in London on Wednesday.
"I think the big thing from the 1922 committee meeting is this - there is nothing that says unity more when you have to get two MPs, from different factions, to come out of the 1922 committee to brief the media and paper over the massive cracks between remain and leave Tory MPs".
Senior Tory Nigel Evans added: 'Bracknell folk voted out by a greater margin than the United Kingdom - we all stood on a manifesto only a year ago to deliver brexit and that is what the people now expect.