However, with the vast majority of both Leave and Remain voters agreeing that the government's negotiating strategy has been a failure, MPs across the aisle have started to have doubts about whether a satisfactory deal will be possible.
But Solicitor General Robert Buckland said while talks may "yield fruit" - he could not guarantee a change in policy. Part B said that if no deal with the European Union was reached by 30 November 2018, the government would have to come to parliament and explain exactly what would happen next.
But, following Ms Cooper's intervention, ministers have agreed to drop a clause stating this could only happen if their family members already in the United Kingdom were over 18 years of age. "I trust our PM to honour the undertaking she gave".
Amid rumours of more departures to follow, he urged fellow Tory MPs to back a push for a "meaningful" vote on any final Brexit deal in the Commons later - and said there should be a second referendum.
If confirmed, the move represents a dramatic climbdown from Mrs May's original plan to offer MPs a "take it or leave it" vote to accept the withdrawal agreement or leave the European Union without a deal.
In such an event, he said Parliament should be able to flex its muscles by requiring ministers to come forward with a plan of action, which MPs would be able to debate and vote on.
Mr Grieve confirmed he voted with the UK Government after receiving assurances, telling Sky News: "I am quite satisfied we are going to get a meaningful vote".
"If it were to turn out there was a problem, we will deal with it".
British Prime Minister Theresa May survived another key Brexit vote on Wednesday (Jun 14) but her pro-European MPs warned they could yet rebel if she backtracks on promises to give parliament a greater say in the final withdrawal deal.
"What it does is put in place a structure if things do go as planned", she said. Now the focus shifts to the price of the rebels' compliance, and it could be a high one for the beleaguered prime minister.
'We must not weaken the PMs hands in her discussions with Brussels so I hope Philip backs Theresa in the lobbies over the next two days despite any personal reservations he may hold'.
She told BBC Radio 4's World At One that "at least half a dozen" junior ministers had been "very uncomfortable for some time" at the Government's direction on Brexit.
He said a concession of this kind would been "revolutionary" as the Commons can not override the government when it came to negotiating global treaties. Within hours of the agreement being struck, each side had a different version of what had been agreed, with the pro-Brexit camp saying the rebels were overstating the scale of the concession.
"There is no inconsistency here". View of Lab MPs is clear, we don't want a hard Brexit'.
The prime minister is said to have agreed a deal with rebels in her Tory party over the "meaningful vote" clause in the EU Withdrawal Bill, according to a person familiar with the negotiations.
In a day of drama, May's position seemed suddenly weaker when junior justice minister Phillip Lee, who has always been critical of the government's Brexit strategy, resigned and said he would vote against the government.
With not all rebels persuaded that May's plan can prevent an economic shock after Brexit, some say they will challenge her plans to leave the customs union again during votes on other bills, on trade and customs, which will be brought back to the house some time before July 24.