The National Audit Office report suggests that the heightened demand for primary school places is partly down to a rising birth rate - the rise in the number of children born in England between 2001 and 2011 was the biggest 10-year increase since the 1950s.
Between 2006/07 and 2011/12, the number of four-year-olds starting reception classes rose by 16%, it says.
It warns that by September 2014, an estimated extra 256,000 primary and secondary school places will be needed to meet the demand. Of these, 240,000 are required in primary schools, with more than a third (37%) needed in London alone.
These extra places are still needed despite a net increase of almost 81,500 places which were created between 2010 and May 2012, and the DfE increasing the amount of funding it gives to local councils to provide spaces.
How many places will be required in the future is uncertain, the NAO says, but it is expected that more will be needed beyond next year.
More top news
The prime minister said "tough new rules" will be introduced to tackle the behaviour of executives who put their workers' pensions at risk.
The musician, whose career spanned six decades and included 18 years with The Kinks, died on Saturday after a fall.
A group of soldiers has broken an Antarctica record by becoming the first all-female team to cross the continent using only muscle power.