But pro-European Tories have accused Mrs May of "caving in" to the party's Eurosceptic wing after the government agreed to their demands to change the wording of the Customs Bill, which is one of the key pieces of Brexit legislation.
MPs voted 307 to 301 to overturn an amendment to the trade bill under which Britain would be forced to join a customs union with the EU if no agreement were reached on frictionless trade by 21 January 2019.
However, it wasn't without drama as junior defence minister Guto Bebb resigned his post.
Government sources confirmed that parliament will, on Tuesday, vote on whether to allow MPs to start their six week leave five days ahead of schedule, according to the Telegraph's Christopher Hope.More news: Justine Greening calls for a fresh vote on Brexit
"The compromise plan that we have from Chequers doesn't achieve any of those".
Tory Remainers claimed the government was being held to ransom by 40 MPs and that Jacob Rees-Mogg appeared to be running the country rather than the Prime Minister. Hammond told MPs: "This does not undermine the bill, it keeps it on the road".
Downing Street insisted the amendments it had accepted were still consistent with the plans in the Chequers agreement but the move may limit the government's room for manoeuvre in exit talks with the EU.
Mr Thomson said the prime minister had previously used major speeches to set out her views on what Brexit actually meant, what she was seeking to achieve and what her red lines were.More news: Trump an 'American patriot' says Turnbull after Putin meeting
"She can not be held hostage by them every time there is a hard vote in the House of Commons - and it now looks as though the Chequers plan is completely falling apart", Ms Blackman said.
Parliament will debate aspects of the Brexit plan later Monday.
May spent the weekend defending her plan, and did so again in a speech at Farnborough airshow Monday, insisting it would protect trade in goods with the European Union and avoid border checks in Ireland. Her authority has been weakened with the resignations of major figures Boris Johnson and David Davis and a series of lesser officials who disagree with her Brexit plan.
Negotiations on the withdrawal agreement and future relationship resumed in Brussels yesterday, with new Brexit Secretary Dominic Raab due to hold his first face-to-face talks with European Union chief negotiator Michel Barnier on Thursday.More news: Britain's May wins parliament vote after bowing to Brexit pressure
On Monday, MP Scott Mann was the ninth Conservative to resign in protest to May's plan.
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