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.
The government subsequently split the bank into two, Northern Rock plc, and Northern Rock (Asset Management), into which was placed its bad debt.
Virgin is buying Northern Rock plc and has pledged no additional compulsory redundancies for at least three years.
The bank currently employs 2,500 people, down from 5,500 when it was nationalised.
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 will get back following the completion of the sale, there is the potential for the Treasury to receive a further £280m over the next few years.
"So on paper, taxpayers end up with a loss of somewhere between £400m and £650m," said our business editor.
The size of the losses contained in the bad bank part of Northern Rock are still uncertain, but could amount to as much as £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 establish a new headquarters in Newcastle, where Northern Rock is based.
It also also agreed not to close any branches and instead to increase their number "as the business' growth allows".
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.
September 2007: The BBC learns the Bank of England has given Northern Rock emergency financial support. Its shares drop 32% in a day and customers queue to withdraw their savings
November 2007: Virgin says it would be interested in taking over the bank
February 2008: Northern Rock is nationalised, the first such case since the 1970s. Chancellor Alistair Darling calls it a "temporary measure"
December 2009: The UK government splits Northern Rock into a "good" bank and "bad" bank, paving the way for a sale of its "good" assets to a third party
June 2011: Chancellor George Osborne confirms that he will sell Northern Rock to a single buyer
November 2011: Northern Rock plc is sold to Virgin Money for an initial £747m.
If Virgin Money sells or lists Northern Rock on the stock exchange in the next five years, it will have to pay the government an additional £50m to £80m.
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.
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.