The change was announced by Theresa May at the Conservative Party conference, and will be implemented as soon as possible following the budget on 29 October 2018. It could deliver up to 10,000 additional homes a year, according to government estimates.
In her speech, May described the UK's shortage of affordable housing as "the biggest domestic policy challenge of our generation".
"It doesn't make sense to stop councils from playing their part in solving it," she said.
In a statement, the government said that interest from councils in areas of high affordability pressure in an additional £1 billion of borrowing made available by the government showed that they were "ready and willing to deliver the homes their communities need". Ending the cap on borrowing would also enable councils to take on smaller projects and sites unsuitable for private development, leading to a more diverse market, it said.
Introduced in 2012, the housing revenue account cap restricts the amount that councils are able to borrow against the value of their existing housing stock.
The announcement was welcomed by the Local Government Association (LGA), which has been campaigning for an end to the borrowing cap.
Housing expert Anne Bowden of Pinsent Masons, the law firm behind Out-Law.com, said that the announcement was "fantastic news for local authorities that have had their hands tied for years by this borrowing cap".
"With the vast majority of authorities pushing up to the limit of what they can borrow, the provision of council housing has been woefully poor for decades," she said.
"The financial freedom to deliver housing is a crowd pleaser but it won't be a silver bullet for the housing crisis. This won't be easy money – there will be a rush of local authorities piling in to boost funding for council housing which could potentially prompt a backlog of approvals for funds. Indeed, boosting finance is just one piece of the jigsaw. Securing or readying appropriate land for an expected influx of new housing and finding skilled workers to build new developments will take time. We have a long way to go before any new council housing schemes are 'shovel ready'," she said.