The government is to set up a fund to give students at risk of dropping out a chance to complete their degree online helped by the Open University.

The prime minister tonight announced the £12m plan to help some of the 35,000 students who drop out every year.

It is a major expansion of the role of the Open University (OU), which this week celebrates its 40th birthday. The university, which has no campus, began life offering late-night lectures on TV in the 1960s, but is now almost entirely web-based. It will partner other universities to design courses students can complete at home.

The Higher Education Statistics Agency reported last month that 7.4% of younger undergraduates left during their first year at university in 2007, up from 7.1% the year before. The Higher Education Funding Council for England said it hoped that up to 15% of the 35,000 who drop out each year could be encouraged to complete their degrees if they could do so at home rather than at university.

Earlier this week, the government announced a separate £20m fund to support online learning.

Gordon Brown, launching the scheme at a Downing Street reception to celebrate the OU's 40th anniversary, said: "The success of distance learning, pioneered 40 years ago by the Open University, has been nothing short of a revolution for higher education. It has opened the doors to a whole new audience of students who have not only seen academic success, but reaped the wider rewards learning brings.

"An approach to higher education that emphasises accessibility and flexibility has put the UK at the forefront of e-learning, and to build on this achievement [we have] a new £20m fund to support centres of excellence for online learning. I am also supporting the Open University with additional funding to further its role as a national leader, working with other institutions, to develop distance learning."

Lord Mandelson, the secretary of state for business, innovation and skills, writing for theguardian.com, said the OU would work with other universities to advise students who were considering dropping out and to develop digital versions of degrees.

Mandelson described the OU as crucial in the expansion of university education. "It was the first step towards a genuine revolution in access to higher education in Britain. Over the last 40 years, more than 2 million people have studied through the OU. There are almost twice as many people enrolled in the OU this year than there were in the entire British higher education system in 1969."

Last week, the head of the National Union of Students, Wes Streeting, argued that lectures had "had their day" and should be replaced with more virtual learning.

The government has already earmarked £225m next year for schemes to encourage students not to drop out of university.

• This article was amended on Thursday 25 June 2009. Students covered by a new government fund will study for their degrees through a range of institutions, not only the Open University, as we said in the article above. What the report should have said was that the fund would give students a chance to complete their degree online helped by the Open University. This has been corrected.