Dr Lee's shock departure came as Brexit Secretary David Davis warned potential Tory rebels that they can not undo the European Union referendum, ahead of a tricky 48 hours in which the Government will try to get its Brexit programme back on track.
May had feared a rebellion by her Conservative MPs over the motion, one of 15 submitted by the unelected upper House of Lords on which lawmakers will vote over the coming two days.
The Lib Dems, who identify strongly as an anti-Brexit party to the point of pushing for a second referendum and running by-elections in Remain-friendly seats on that platform, opposed the Government.
"The main reason for my taking this decision now is the Brexit process and the government's wish to limit parliament's role in contributing to the final outcome in a vote that takes place today", Lee, who voted to remain in the European Union during Britain's 2016 referendum, said on his website.
Instead, the government has proposed reporting its efforts to secure a customs arrangement.
Given that after November 30, the House of Commons looks set to be empowered in the negotiations, it would not be in Barnier's interest to negotiate a harder form of Brexit before the U.K.'s self-imposed deadline.
The meaningful vote is probably the most unsafe of the Lords amendments to the EU (Withdrawal) Bill - because it tees up an unpredictable vote on the final terms of Brexit, towards the end of this year, and opens up the possibility that MPs could demand that ministers change policy, in the event the terms were rejected by the House, or no deal was reached in the talks with the EU.... they could even demand (drumroll) a second referendum...
A minister long critical of the government's handling of Brexit resigned earlier in protest at what he called its "wish to limit" the role of lawmakers in the process. Lee, who voted for Britain to remain in the E.U.in the 2016 referendum, said in a statement he was "incredibly sad" to resign but did so in order to vote against the government's position on a key amendment to the bill.
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"But if the Lords amendments are allowed to stand, that negotiating position will be undermined".
MPs will on Wednesday debate a series of other amendments to the bill, which if passed, would seek to force May to change course and negotiate to stay inside a customs union and the single market after Brexit.
Parliamentary debates about complex legal amendments rarely rouse much heat, but passions run high over anything to do with Brexit.
The Sun had a frontpage of British icons including Stonehenge, a fish and chip shop, a London bus and a football, saying "Great Britain or Great Betrayal". The Daily Express featured the British flag as its front page with the headline: "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.
In the tense atmosphere where it was not clear which way the vote would go, the government secured its victory only after offering concessions to one of the leaders of a group of Conservative lawmakers who were threatening to vote against May.
In fact, her party is far from united. May's Cabinet is divided between ministers including Foreign Secretary Boris Johnson, who support a clean break with the European Union, and those such as Treasury chief Philip Hammond who want to keep close ties with the bloc, Britain's biggest trading partner.
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.