The Co-operative Group has agreed terms to buy 632 branches from Lloyds Banking Group in a deal that could create a new force in UK High Street banking.
About 4.8 million Lloyds customers will be transferred to the Co-op, giving it about 7% of the current account market.
The Co-op will pay £350m upfront and up to an additional £400m based on the performance of the combined business.
The sale was demanded by European regulators after Lloyds was part-nationalised in 2009.
After buying HBOS at the height of the financial crisis in 2008, Lloyds then needed a government bailout, and is still 40%-owned by the taxpayer.
The deal with the Co-op will take the Co-op's total branches to almost 1,000.
The price tag, of up to £750m, is much less than the £1.5bn that had originally been expected for the sale.
"Those were the numbers that were speculated 12 months ago but during the last 12 months the business that we're buying has changed significantly, [and] we think we're paying a very fair price," Co-op chief executive Peter Marks told the BBC.
Under the deal, Lloyds is also providing £1.5bn of capital for the branches, meaning it faces a loss of about £750m on the sale.
BBC business editor Robert Peston said it looked like "an amazingly good deal" for the Co-op.
The sale, which is still subject to approval by the Financial Services Authority, is expected to be completed by the end of November 2013.
The Chancellor, George Osborne, welcomed the deal.
"This is another step towards creating a new banking system for Britain that gives real choice to customers and supports the economy," he said.
"The sale of hundreds of Lloyds branches to the Co-operative creates a new challenger bank and promotes mutuals. This follows the sale of Northern Rock to Virgin Money in January and represents another important step towards a more competitive banking sector."
'Bring back trust'
Lloyds group chief executive Antonio Horta-Osorio said: "In agreeing to move ahead with the Co-operative we provide greater certainty for our customers and for our shareholders.
"In addition to an upfront consideration, we will also get to share in the future financial performance of the combined banking business which will be an effective challenger with a strong customer focus."
Lloyds said that the branches being sold would be rebranded to TSB next summer and would transfer to the Co-op under that brand.
They include all Cheltenham & Gloucester branches, and all Lloyds TSB branches in Scotland.
Co-op chief Peter Marks said the deal was good news for the industry.
"What the banking industry needs is to bring back trust. We've seen over the last few years, and particularly over the last few weeks, trust has deteriorated in big banks," he said.
"This represents a major change to the face of retail banking in the UK."
He added that the Co-op, which is owned by its members, was a very different bank to the main High Street banks.