Northern Rock is being sold to Virgin Money for £747m, the government has announced.
The bank was nationalised in 2008 following its near collapse at the onset of the global credit crunch.
Northern Rock plc will be rebranded as Virgin Money, which has pledged no compulsory job cuts for three years.
BBC business editor Robert Peston said the sale would see taxpayers end up with a "paper" loss of somewhere between £400m and £650m.
The bank currently employs 2,500 people, down from 5,500 when it was nationalised.
On nationalisation, the government subsequently split the bank into two, Northern Rock plc, and Northern Rock (Asset Management), into which was placed its bad debt.
Sources at Northern Rock told the BBC that there were cheers at the bank's Newcastle headquarters when the news of the Virgin Money deal was announced.
The government said Northern Rock customers would see no change to their accounts and services and would not need to take any action.
BBC business editor Robert Peston said taxpayers had injected £1.4bn into Northern Rock plc.
He added that, in addition to the immediate £747m the government would get back following the completion of the sale, there was the potential for the Treasury to receive a further £280m over the next few years.
The size of the potential losses contained in the bad bank part of Northern Rock is still uncertain and it still owes the Treasury £21bn.
Chancellor George Osborne said: "The sale of Northern Rock to Virgin Money is an important first step in getting the British taxpayer out of the business of owning banks.
"It represents value for money, will increase choice on the High Street for customers, and safeguards jobs in the North East."
The sale of Northern Rock plc is expected to be completed on 1 January 2012.
The government said it had no plans to sell Northern Rock (Asset Management).
Virgin Money has pledged to maintain its operational headquarters in Newcastle, where Northern Rock is based.
It has also agreed not to close any branches and instead to increase their number "as the business' growth allows", and support Northern Rock's charitable foundation for a year.
In addition to paying £747m on completion of the sale, the government said Virgin Money was "expected" to pay an additional £50m within six months, and then a further £150m.
If Virgin Money sells or lists the combined business on the stock exchange in the next five years, it will have to pay the government an additional £50m to £80m.
Virgin Money chief executive Jayne-Anne Gadhia confirmed to the BBC that the bank intended to float its shares on the stock market within two to five years.
She added: "The great thing about this business combination is that the two businesses lock together very well.
"Virgin Money has credit cards, insurances and investments, and Northern Rock has mortgages, savings and current accounts.
"There is no need for any jobs to be merged together, in fact this is a story for growth."
'Message of confidence'
Ron Sandler, Northern Rock executive chairman, said: "The return of Northern Rock to the private sector has always been one of our key objectives.
"We said that this would be done at the right time and when there was a proposition in the best interests of taxpayers and other stakeholders.
"It is a very positive outcome for the company following a significant restructuring process."
The Unite trade union said it hoped the announcement of the sale to Virgin Money "will be the start of a secure future" for Northern Rock's workforce.
The leader of Newcastle City Council, Nick Forbes, said it was "delighted that the future of Northern Rock has now been decided".
"The decision by Virgin Money to make Newcastle their home sends a message of confidence in our city and the wider North East," he said.
Adrian Coles, director general of the Building Societies Association, said the announcement of Northern Rock's sale was a "bittersweet moment".
"On the plus side, de-nationalising this bank is a positive step. However, we would have welcomed Northern Rock's return to the mutual sector after 14 years' absence," he said.
"Northern Rock had over 100 years as a successful building society, but only 10 years as a plc bank before the queues formed outside its branches."
A savings and mortgage bank, Northern Rock currently has more than 70 branches.
The bank was formed in 1997 when the former Northern Rock Building Society floated on the London Stock Exchange.
It was delisted following its nationalisation.
Virgin Money had approached the Treasury about making a bid for Northern Rock back in 2008, but it was rejected.
Virgin Money is part of Sir Richard Branson's Virgin Group.