May is due to address Parliament Wednesday, opening five days of debate before a December 11 vote on the divorce agreement.
MPs will decide whether to reject the terms of the UK's withdrawal and future relations with the European Union on Tuesday 11 December.
The House of Commons lawmakers voted 311 to 293, backing the ruling that the government was in contempt of the Parliamentary procedure.
Andrea Leadsom, Leader of the House of Commons, said on Tuesday that this had been a "full and frank exposition", and that releasing the full advice would set a risky precedent.
Mr Bercow made the comments in response to demands from Labour, the DUP and four other opposition parties, which had complained that the summary legal advice released by the Government on Monday did not comply with a Commons resolution passed in November.
Monday's angry session of Parliament did not bode well for the December 11 vote, which will come at the end of five days of bruising debate starting on Tuesday.
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Speaking in the Commons after the vote, Leader of the Commons Andrea Leadsom said: "We've tested the opinion of the House twice on this very serious subject".
The government had initially lost a vote, 311 votes to 307, on their own amendment which would have meant the case of "contempt" being referred to the Parliamentary standards committee.
The Prime Minister told Cabinet that the publication of the document setting out the legal position on Monday and the Attorney General's appearance in the Commons "are by themselves extraordinary steps for any government to take".
At the Commons Business Committee, Toyota Europe deputy managing director Tony Walker warned that without a deal to protect cross-continent supply chains, its operations in the United Kingdom would face major challenges.
Theresa May said the United Kingdom would enjoy a better future outside the European Union.
"The British people want us to get on with a deal that honours the referendum and allows us to come together again as a country, whichever way we voted", May will say, according to comments released by her Downing Street office.
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Some 26 Tory MPs - including former ministers Sir Michael Fallon, Damian Green and Sir Oliver Letwin - rebelled on the amendment tabled by ex-attorney general Dominic Grieve.
That result - which could prevent a "no deal" Brexit - saw the pound rise back to where it started the day, rounding off a rollercoaster day for sterling.
"It's a false choice to say it's the PM's deal or chaos", said Conservative lawmaker Sam Gyimah, who quit as a minister on Friday over May's deal.
She said her agreement respected the outcome of the 2016 referendum while protecting vital trading relations with the European Union and insisted the British public wanted closure on the issue. Hard-liners, who support a clean break, argue that the agreement forces the United Kingdom to remain in a customs arrangement with the European Union without a clear end date.
She also said Parliament would be consulted on the government's negotiating mandate as it embarks on talks on the UK's future relationship with the EU.
But the deal has done little more than boost opposition at the hardline edges of the debate. "That is the nature of a negotiation", she said.
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And she will insist: "This is the deal that delivers for the British people".