The Prime Minister, Tony Blair, and the Chancellor, Gordon Brown, had been expected to block Liberal Democrat moves to give the Bill, and another on energy efficiency, as much parliamentary time as they needed.
The Bill to license minicabs in the capital, put forward by Sir George Young, a former Tory transport minister, was blocked by a single MP - Eric Forth (Con, Bromley and Chislehurst), a campaigner against bureaucracy.
He also blocked a Bill that would require mortgage lenders to carry out energy surveys.
In response, the Liberal Democrats put forward a scheme to rescue the Bills, by giving up one of the few days they can set Commons business to debate the issue.
Mr Blair, Mr Brown, John Prescott, the Deputy Prime Minister, Robin Cook, the Foreign Secretary, Jack Straw, the Home Secretary, and Ann Taylor put their names to an amendment saying "the existing procedure and time available for private member's bills should not be altered". Such proposals should be examined by a select committee, it said.
However, in a surprise U-turn in the Commons, Mrs Taylor, the Leader of the Commons, said the Government would have no difficulty reaffirming its support for the two Bills.
She said the Government was happy to be associated with this legislation, which it had helped draft.
"If members acted co- operatively there could well be time for several of those Bills to complete their passage through this House on that day [3 July - when the matter comes up for debate again] and proceed to the House of Lords.
"Unfortunately, I think there may be one or two members who may still be intending to object to those Bills on 3 July, but I think it is important they know how strong the feeling is in the rest of the House."
Following the Government's decision not to oppose the amendment, the Liberal Democrat proposal was passed without objection.
Urging the Government to accept the Liberal Democrats' amendment, Paul Tyler (Cornwall North) said that the Bills had considerable cross-party support, including the backing of the Prime Minister.
He added that they also had very strong public support, the legislation had been fully considered in committee and was totally uncontroversial.
"All parts of the House, all members, of any party and of no party, do have a responsibility to reflect public concern about the way that we handle our business here, and that includes public members' bills."
The changes will significantly affect the lives of millions of people for the better, the spokesman Jackie Ballard (Lib Dem, Taunton) told the Commons.
It is at present illegal for an employer to conduct a full criminal check on records of drivers. Last year, there were 67 people assaulted and 18 raped in minicabs in London.
Ms Ballard pointed out that there had been no credible opposition to the Bill, "except for the reaction of one honourable member [Mr Forth], who I note is not in his place today."Reuse content