Michael O'Brien, the Immigration minister, stressed that the Home Secretary, Jack Straw, had proposed increasing the cash payments made to asylum-seekers and pledged that the speeding-up of case handling would be brought forward by a year to next year.
However, Neil Gerrard, the Labour MP for Walthamstow and chairman of the all-party refugee group, warned during the Bill's report stage that ministers could still face a minor revolt today when the legislation returns for its Third Reading.
But he admitted that Mr Straw's concessions last week had won over most of the 45 or so backbenchers who had earlier supported his amendment to reverse the plan to introduce vouchers instead of cash benefits for asylum- seekers.
"There has been a general view that while people are not entirely happy, there have been concessions, unlike over welfare reform, and we should recognise that and not discourage the Government from adopting such an attitude," he said.
Under original proposals the cash benefits for adults seeking asylum would have been limited to pounds 1 a day and 50p for their children, with vouchers for food. Mr Straw announced last week this would be increased to pounds 10 a week for adults and children, raising the cash support for a family of four to pounds 40 with pounds 50 in vouchers.
Although this recast the financial support without raising the overall cost, Mr Straw also agreed to put on a fast track the system for processing the asylum claims by families, to ensure that by April next year the average delay will be reduced to two months.
Backbenchers contrasted Mr Straw's willingness to listen to them with the refusal to compromise by the Social Security Secretary, Alistair Darling, over disability benefits, which provoked 67 backbenchers to rebel last month.
But Mr Gerrard warned that some Labour MPs were still planning to rebel and vote against the Government when the Bill had its Third Reading tonight.
Earlier, MPs debated the Government's decision to "guillotine" debate on the Bill, giving the Commons just a few hours last night and today to complete all remaining stages.
As MPs were debating the issue, pressure groups handed into Downing Street a petition signed by more than 100 organisations.
Victims of Rape said the Bill's proposals would have a devastating impact on women and children fleeing abuse.
Helen Bamber, of the Medical Foundation for the Care of Victims of Torture, said: "The proposed food voucher scheme is dehumanising while the policy of `no choice' dispersal of asylum-seekers around the country is likely to cause widespread misery."
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