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.
On Tuesday, British House of Commons voted 311 to 293 to hold May's government in contempt.
Her spokesman said the cabinet had discussed the motion on Tuesday but maintained that ministers must be able to obtain candid legal advice "without fear that it will be immediately published".
Attorney General Geoffrey Cox insisted the Government had "gone out of its way" to satisfy the terms of the humble address to the Queen passed by Parliament on November 13.
A judge in the European Union ruled on Tuesday before this vote took place that the United Kingdom could cancel its Brexit plans without getting the approval of all the remaining EU member states. But the contempt vote demonstrated the fragility of May's government, which does not have a majority in Parliament.
Prime Minister Theresa May's official spokesman told reporters in Westminster that "this is not a final judgment", adding: "It does nothing in any event to change the clear position of the Government that Article 50 is not going to be revoked".
And MPs backed a motion giving the Commons a direct say in what happens if her deal is rejected next Tuesday.
May, for her part, has said the full extent of advice received by her government over the Brexit deal is confidential under lawyer-client privilege.
The government lost the critical vote by 311 votes to 293, a stinging defeat for May at the beginning of a week of votes on issues related to Brexit.
The amendment isn't legally binding - but it carries huge political weight and would be very hard for the government to ignore.
An attempt by ministers to refer the whole issue, including the government's conduct, to the committee of MPs was earlier defeated by four votes.
The fate of those will determine if her Brexit deal succeeds, whether the United Kingdom could be headed for no deal, a second referendum, or even a general election.
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".
Advocate General Manuel Campos Sanchez-Bordona told the European Court of Justice that EU law "allows the unilateral revocation of the notification of the intention to withdraw from the EU".
It means that instead of the government having to come back to tell MPs what their next steps are - and MPs voting on that - they would theoretically be able to vote on what they wanted the government to do as well.
Soft-liners are against the deal because they believe that it won't secure the closest possible British-EU political and economic ties. To complicate matters even more, between 50 and 80 hard-line Conservative members of Parliament are likely to reject the deal.
If, against the odds, May wins the vote, Britain will leave the European Union on March 29 on terms negotiated with Brussels - its biggest shift in trade and foreign policy for more than 40 years.