It has also intensified pressure on a prime minister who lost her party's parliamentary majority at an ill-judged election previous year and tested her already weakened authority.
The British government needs to move on from internal debates and make choices about what it wants from Brexit, and must acknowledge that it can not have its cake and eat it, Irish Prime Minister Leo Varadkar said on Wednesday.
The concession on a meaningful vote came after intensive horse-trading on the floor of the House of Commons, with chief whip Julian Smith shuttling between Tory backbenchers during debate on Lords amendments to the EU Withdrawal Bill.
Defeat on the bill would cause a leadership crisis for May.
Theresa May has been warned "it's crunch-time" after a last-minute change was made to a critical amendment to the Brexit bill after the government effectively turned it into a confidence vote.
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".
Under the Government's plan, MPs will only be able to debate the Prime Minister's next move if no deal is struck or they vote down the deal, and would have no power to direct what should happen next. Instead, MPs will be offered a vote on an unamendable statement, which would effectively turn any vote into to a vote of confidence in the prime minister.
The rebels thought they might reach agreement on the amendment last night, but this morning said they were still waiting.
"I've been through this before when in opposition and now that when we're in Government, because if the House makes the concession of allowing the dialogue to continue and I can see the merit of that happening, it has got to be done in good faith".
"Grateful for the conversations but without consultation what was agreed earlier today has been changed".
The so-called "meaningful say" vote was defeated by 324 votes to 298.
"If, in the future, I am to look my children in the eye and honestly say that I did my best for them I can not, in all good conscience, support how our country's current exit from the European Union looks set to be delivered", said Dr Lee.
An agreement that defused a potential rebellion over handing parliament more control over Britain's exit from the European Union looked in danger of unraveling on Wednesday, when the two camps argued over the shape of a possible compromise on a "meaningful vote".
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.
However, May's decision to refuse to offer MPs a veto means the future of her Brexit legislation and her authority as prime minister are now once again back in doubt. Britain and the EU agree there must be no customs posts or other border infrastructure to impede the free flow of people and goods, but the United Kingdom has not said how that can be achieved if it is outside the customs union.
The pro-independence party accuses the British government of trying to seize powers that will be handed back from Brussels after Brexit and which the SNP believes should go to Scotland's Edinburgh-based parliament.
NY sues Trump Foundation for 'illegal conduct'
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