The Prime Minister blocked moves to force oversubscribed schools to take an equal number of pupils from each ability range, fearing it would be portrayed as social engineering and an attack on school autonomy. Early drafts of the white paper said that the practice, known as "fair banding", should be compulsory.
But despite support for the policy by senior figures within the Department for Education and Skills and advisers, the final version promised only to encourage schools to adopt the policy.
John Bangs, the National Union of Teachers' head of education, said: "This will not stop schools creaming off pupils they want."
Ministers hope the white paper will revolutionise the school system, extend parental choice through new "trust" schools and tackle underperforming schools with tougher inspections.
It signals the death of LEA-run community schools in England. All new schools must now be either foundation schools, voluntary aided, or academies - or trust schools free from council control, run by businesses, churches, parents or voluntary groups.
The Welsh Assembly government has rejected trust schools and reiterated its pledge to community comprehensive schooling.
A spokesperson said: "This is the right solution for Wales and one that enjoys the overwhelming support of our schools, LEAs, teachers and parents.
We are sustained by evidence from Estyn, our school inspectorate, that the standards achieved by pupils in Wales continues to rise."
The DfES proposals should, however, lead to new powers being devolved to the Welsh Assembly, particularly for setting standards for school meals.
Opponents of the paper, including senior Labour figures and backbenchers, fear the trust schools will become elite centres for the middle classes.
A Labour whip told The TES: "There are large parts of the white paper that present lots of problems for many MPs. They agree with John Prescott that the choice will only really be for middle-class parents. The Government will have to do a lot to convince us that it will not be the case."
The paper, Higher Standards, Better Schools for All, sets out a new duty for local authorities in England to promote "choice, diversity and fair admissions" as part of a radical shift from being providers to commissioners of services.
Councils will be unable to create any new community schools, and existing ones will be urged to opt out of local authority control.
Ruth Kelly, Education Secretary, said parents and heads would be eager for schools to work with charitable trusts because they would have more freedom.
However, the trust schools will have to obey the national codes on admissions and will only be able to deviate from the national curriculum and rules on pay and conditions for their staff if they get permission from the Government.
The white paper also suggests giving poorer families "choice advisers", and guaranteeing free transport for poorer pupils to their three nearest schools.
Heads' leaders said there was little in the proposals for schools. John Dunford, general secretary of the Secondary Heads Association, said: "It will be ignored by headteachers." The white paper will form the basis of a forthcoming education bill.
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Higher Standards, Better Schools for All is at www.dfes.gov.ukpublications