In a statement, Northern Rock said the government will receive £747m in cash on closing of the sale, expected to complete by the end of the year, with the potential in the future to receive over 1bn pounds in total.
The operational headquarters of the combined business will be based in Newcastle.
Virgin Money has committed to no further compulsory redundancies beyond those already announced by Northern Rock, for at least three years
The total number of Northern Rock branches will be retained and extended in due course as business growth allows.
The former Northern Rock business was taken into temporary public ownership in February 2008. Following the restructure of the former business, the new Northern Rock plc was created as a savings and mortgage bank, on 1 January 2010.
One of the key conditions of state aid was the return of the company to private ownership.
The whole enlarged group will be branded Virgin Money following completion of the sale.
For customers, it is business as usual, Northern Rock said.
Ron Sandler, executive chairman of Northern Rock plc said the deal “delivered value to taxpayers”.
“Today’s announcement demonstrates the enormous progress that has been made at Northern Rock, which is great testimony to the commitment and spirit of the people who work here.
“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.”