The defeat was widely expected, but the scale of the House of Commons' vote - 432 votes against the government and 202 in support - was devastating for May's fragile leadership.
Labour Party leader Jeremy Corbyn immediately proposed a vote of no confidence in May's government, which could force her from office or compel a new national election.
"The voting result in the British House of Commons was very clear, " Mr Luik said.
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Labour says further no-confidence votes could follow if this one fails.
But senior Labour figures accept it is not likely to succeed, as she has the backing of Tory rebels and the DUP's 10 MPs, who less than 24 hours ago helped inflict a humiliating defeat on her. For the first time, May said she is willing to talk to the leaders of all opposition parties to reach a deal.
Foreign Minister Heiko Maas said delaying Brexit beyond March 29 would make no sense.
May promised to consult lawmakers on future moves, but gave little indication of what she plans to do next.
European Union leaders were pleading Wednesday with the U.K.to finally get its act together on Brexit, end internal strife and come up with a realistic plan to leave the bloc - well over two years after Britain made a decision to leave the EU and with its departure scheduled in less than 10 weeks.
"If you can't resolve the impasse here in Westminster, than you have to refer it back to the people", said Labour Party lawmaker Chuka Umunna, who supports a second referendum. And scores of Mrs May's own Conservative MPs have said they will vote against the government, majority because they think the deal would leave Britain too close to the European Union.
All agree that it's for the United Kingdom government to decide on the next steps.
Germany and France are said to be willing to extend withdrawal negotiations until 2020, The Times of London reports: "Previous planning had centred on a three-month delay to Brexit from March 29 until the end of June but now, according to multiple sources, European Union officials are investigating legal routes to postpone Britain's withdrawal until next year".
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The most contentious section of the deal is an insurance policy known as the "backstop" that is created to prevent the reintroduction of border controls between the U.K.'s Northern Ireland and European Union member Ireland.
"Were the backstop to enter into force in whole or in part, it is meant to apply only temporarily, unless and until it is superseded by a subsequent agreement", Tusk and Juncker added, pledging the "necessary political impetus and resources" to quickly reach that more permanent agreement. With just over two months to go until the scheduled departure, Britain is still undecide on what to do.
Brexiteers banged drums and rang a "liberty" bell, while pro-EU demonstrators handed out anti-Brexit stickers in Parliament Square beside two huge video screens set up for the live broadcast of the final speeches and the vote.
One group waved blue-and-yellow European Union flags, the other brandished "Leave Means Leave" placards.
Even the most optimistic members of May's inner circle didn't think the government could win the so-called "meaningful vote".
The Guardian's view on May's Brexit deal is that it is "dead before the vote", reflecting the popular consensus that it had no hope of being endorsed by MPs.
She told lawmakers that the choice was plain: support her imperfect compromise deal - and the only one that Europe will abide, she stressed - or face the cliff edge of a no-deal Brexit.
But May urged lawmakers to listen to the British citizens who voted to leave the EU.
"Time is nearly up", he said.
"Every business will feel no-deal is hurtling closer", said Carolyn Fairbairn, director-general of the Confederation of British Industry.