The government lost the vote by 321 votes to 278, forcing Mrs May to give MPs a vote on delaying Brexit tonight.
After Wednesday's vote, they said that while it was one thing for the British government to reject a so-called no-deal exit, at some point it would have to figure out the alternative, a deal that could actually pass.
Labour MP Ruth Smeeth has resigned as an aide to deputy leader Tom Watson "in order to vote against a second referendum".
She said parliament was now at an impasse: "Does it wish to revoke Article 50 (announcing intention to leave the EU)?"
"It remains to be seen if there are enough votes in parliament for another referendum" or to reverse Brexit, he told AFP: "So far it doesn't seem that there are".
"The Prime Minister has run down the clock and the clock has been run out on her", he said.
However, the PM warned if the deal - which has already been twice rejected by overwhelming majorities - is not approved, a longer extension will be needed, requiring Britain to take part in the European Parliament elections in May.
"It has been completely consistent with Government policy to date that we should not leave on March 29 without a deal".
"On the European Union side, we have done all that is possible to reach an agreement", he said, adding: "It is hard to see what more we can do".
Newly Independent MP Anna Soubry said Labour had betrayed its voters by not supporting the amendment
The pound tumbled on Tuesday after the United Kingdom government's top legal advisor cast doubt on Prime Minister Theresa May's last-gasp changes to her Brexit deal hours before a vital vote that few think May can win.
A second, but much less likely, possibility is that she fails to get her deal through parliament next week and then herself decides that she prefers a no-deal Brexit to a potentially indefinite delay.
The party was criticised for the move by some supporters of another referendum, but the official People's Vote campaign issued a statement that read: "We do not think today is the right time to test the will of the House on the case for a new public vote". "We need a constructive and positive vote".
Even if the miracle happens, as early as March 20 Mrs May will be asking for a short extension until June 30 to clear a Brexit legislative timetable that has been delayed amid the Westminster chaos.
Since then, the value of the British pound has plummeted, and companies have begun making contingency plans for a post-Brexit world in which European workers can not easily move to Britain, tariffs might be resurrected and trade becomes harder.
They'd say 'What do you mean, you're going to take another vote?' So that would be tough.
Last night, Members of Parliament (MPs) voted by 312 to 308 to reject a no-deal Brexit, which sent the pound soaring to over 2%.
The UK would have to provide "credible justification" if lawmakers voted for a deadline extension, he said.
The EU has done "all that is possible to reach an agreement" and the solution to the deadlock "can only be found in London". We will keep up the pressure for a People's Vote on the final Brexit deal.
Nigeria's Airline to Keep Boeing 737 Deal
That delivered a one-two punch for an aircraft type that has ranked among the world's safest since it started flying in the 1960s. The UK Civil Aviation Authority (CAA) announced Tuesday it was grounding all 737 MAX aeroplanes from UK airspace.