The government must start talks with a group of pro-EU Conservative lawmakers to find a new wording for its position on giving parliament a "meaningful vote" on a Brexit deal.
He said: '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 exit from the European Union looks set to be delivered'.
Earlier, Brexit minister David Davis told parliament a defeat would undermine negotiations with Brussels and warned lawmakers the government would never allow them to "reverse Brexit".
The government has also put down some amendments aimed at uniting the party around a compromise.
"When the Government is able to set out an achievable, clearly defined path, one that has been properly considered, whose implications have been foreseen, and that is rooted in reality and evidence, not dreams and dogma, it should go to the people, once again, to seek their confirmation", he wrote.
A move that would have given MPs the power to stop Britain leaving the European Union (EU) without a deal was rejected in the House of Commons.
Lee said that within government he "found it virtually impossible to help bring sufficient change to the course on which we are bound".
"This Government is delivering on the decision made by the country in the referendum to leave the European Union and we will not accept anything that prevents us from taking back control of our money, laws and borders". "And I can not bring myself to vote for it in the bastion of liberty, freedom and human rights that is our Parliament".
He added: 'If Brexit is worth doing, then it is certainly worth doing well; regardless of how long that takes'. "It is, however, irresponsible to proceed as we are".
Fellow potential rebel Heidi Allen also said Downing Street's position contradicted what the prime minister had told backbenchers in her parliamentary office on Tuesday.
Liberal Democrat Brexit spokesman Tom Brake said: "As has become a tradition in Brexit negotiations, the Tories have been forced to cobble together a compromise".
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Many politicians echoed the Obama administration's reasons for joining the council when speaking out against the decision. And the move is also unprecedented, because the U.S.is the first country to leave the council without being forced out.
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.
A section of Labour MPs are expected to defy the official party position and vote in favor of a Lords amendment to keep the United Kingdom in a Norway-style trading arrangement, better known as the European Economic Area (EEA), post-Brexit.
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". If and when agreement can be reached, the new amendment will be introduced in the House of Lords, when the bill returns there in the next stage of its passage through parliament.
The government won the first set of votes Tuesday, but looked set to face defeat on the issue of whether Parliament should have a "meaningful vote" on the Brexit deal.
Ahead of the 1922 meeting, Conservative Remain campaigner Sarah Wollaston called for "further concessions" on the customs union.
Dismissing the Government's compromise, she tweeted: "Merely issuing a statement in response would make it a meaningless final vote".
Grieve's amendment was in three parts.
The two sides aren't yet clear exactly on the terms of their deal, but it looks likely that Parliament will emerge with a greater say in the process.
The EEA mirrors the single market, meaning Britain would have to abide by the free movement of people and other rules - an outcome pro-Brexit MPs argues defies the will of the people.
A separate vote against the Lords customs union amendment was rejected by 325 votes to 298, at a majority of 27.