Justice minister Phillip Lee is the first minister to resign over the government's Brexit policy.
Lee said "the people, economy and culture of my constituency will be affected negatively" by Britain's European Union departure, and it is "irresponsible to proceed as we are". It also increases the prospect of MPs forcing a referendum on the terms of the eventual deal or even of a snap election before the end of the year.
The Bracknell MP, who called for a second referendum on whatever deal Mrs May secures from the European Union, later told the Commons there was growing evidence that the Government's Brexit policy is "detrimental to the people we were elected to serve".
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.
May objected to the amendment - inserted by the House of Lords - because she said it would tie her hands in the negotiation.
But Tory MP Andrew Bridgen, a leading Brexit backer, said the concessions could "come back to haunt" the government if they amounted to a veto over the terms of the UK's departure.
Pro-EU demonstrators hold placards and wave flags as they protest against Brexit, outside of the Houses of Parliament in central London on June 11, 2018.
A junior British justice minister said he will resign and vote with pro-European Union rebels who want parliament to have the power to force the government to go back to the negotiating table if they reject a Brexit deal.
An hour before the vote, the government's solicitor general, Robert Buckland, promised lawmaker Dominic Grieve talks on increasing the powers of parliament if May was unable to reach agreement in Brussels.
"This needs to be resolved", Andrew Bridgen, a pro-Brexit Conservative lawmaker, told Reuters.
"It's not practical, it's not desirable and it's not appropriate", he said.
Instead, the government has proposed reporting its efforts to secure a customs arrangement.
Parliamentary debates about complex legal amendments rarely rouse much heat, but passions run high over anything to do with Brexit. The Daily Express thundered: "Ignore the will of the people at your peril".
Phillip Lee, who resigned this morning, gave an impassioned speech from the "naughty corner" on the backbenches - flanked by Remainers including Bob Neill, Nicky Morgan, receiving congratulations for his decision by Soubry and Sarah Wollaston.
Opening the debate, Brexit Secretary David Davis insisted the government would abide by three principles to defend the will of the British people.
No - Even with the support of 10 Democratic Unionist Party MPs, May has a working majority of just 13 in the Commons, which means she can be defeated by a rebellion of as few as seven Tory backbenchers.
But once Britain leaves the E.U.it must find a way to regulate or otherwise account for goods crossing its border into Northern Ireland while keeping it open. But May is facing a potential rebellion from some Conservative lawmakers who want to retain close ties with the bloc after the United Kingdom leaves in March 2019.
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.