The vote comes after Theresa May and left-wing Labour Party leader Jeremy Corbyn sought a compromise deal to prevent an abrupt British departure from the European Union.
The leaders of the EU's 27 remaining countries gave the United Kingdom until April 12 to leave the bloc or to come up with a new plan, after lawmakers thrice rejected an agreement struck between the bloc and May.
"It is very disappointing the cabinet has made a decision to entrust the final handling of Brexit to Jeremy Corbyn and the Labour Party", said Boris Johnson, whose sentiments were further echoed by his fellow Conservative Party member Jacob Rees-Mogg, who added, "What was announced. was an attempt to overturn Brexit in an attempt to do a deal with socialists". "The meeting was useful but inconclusive". He himself has said such a vote should be restricted to specific circumstances.
"I simply can not support any further extension to Auricle 50, and this obviously means I can not stay in Government".
Talks will continue on Thursday.
It followed a Cabinet session that lasted more than seven hours.
May also said she would ask the European Union for a further delay to Britain's departure date - postponed once already - to avert a chaotic and economically damaging no-deal Brexit on April 12.
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More than 400,000 Israelis live in West Bank settlements, while a further 200,000 live in settlements in occupied east Jerusalem . Tank driver Baumel and two other soldiers, Zvi Feldman and Yehuda Katz, had been listed as missing and presumed dead.
European Union leaders may hold a special summit on April 10 to consider any request from Britain for a longer delay to Brexit - or to make last-minute preparations for Britain's death without a deal two days later.
As it stands, Britain will still leave the European Union on April 12 without a deal, something many Conservative lawmakers would like to happen but a scenario businesses fear could wreak chaos and cause huge economic damage.
"If we cannot agree on a single unified approach, then we would instead agree a number of options for the Future Relationship that we could put to the House in a series of votes to determine which course to pursue", said May.
She crucially added that she was now willing to bend her previous principles and would listen to proposals for much closer post-Brexit relations with the European Union than most in her Conservative party are ready to accept.
A source said ministers at the meeting were split 14-10 against asking for a long extension to the Brexit process.
Adams, in his letter, accused the prime minister of "cooking up" a deal with Corbyn "who has never once in his political life put British interests first".
However, Brexiteers believe it would not amount to the Brexit that had been promised to people in the referendum because it would not leave Britain free to make its own trade deals with the rest of the world on its own terms.
Using a nickname that plays on May's reputed robotic inflexibility, one Brexit-supporting Conservative lawmaker told Reuters: "The Maybot has gone haywire - we've got to find the "off" switch".