The banks, which had until 5pm on Monday to strike an agreement, said the combined group would “create the UK’s first true national competitor to the large incumbent banks” with more than six million customers.
It would also see the group’s operations come under the Virgin Money brand in a licensing deal with Virgin Enterprises, the announcement said.
Under the terms of the all-share deal each Virgin Money share would be exchanged for 1.2125 new CYBG shares.
The companies said it represented a premium of 19% to the closing price for Virgin shares on 4 May, at the start of the offer period.
The new share structure would see Virgin investors own approximately 38% of the combined group.
But a union said it was demanding urgent meetings with management amid warnings by CYBG that as many as 1,500 jobs were likely to be lost among the combined group’s current workforce of 9,500 as duplication and other cost synergies are identified.
It is believed management roles would be worst hit across the combined group, which is to have its headquarters in Glasgow.
CYBG chairman Jim Pettigrew, chief executive David Duffy and finance chief Ian Smith would all remain in their roles.
Virgin Money’s chief executive, Jayne-Anne Gadhia, would not leave completely, the statement said, suggesting she had “agreed in principle” to serve in a consultancy role as a senior adviser to Mr Duffy.
She said of the deal: “The combination of Virgin Money with CYBG will have greater scale to challenge the big banks.
“It will also accelerate the delivery of our strategic objectives, particularly the expansion of the products we offer to
She added: “I am especially pleased that we have received a number of important commitments from CYBG.
“We have obtained assurances from CYBG regarding our employees (including a commitment to leverage the best talent from both CYBG and Virgin Money) and our Gosforth headquarters.
“The combined group will remain a committed voice behind the Women in Finance Charter as well as working to reduce the gender pay gap.
“This is a compelling deal for our shareholders, that accelerates value delivery and represents the beginning of the next chapter of the Virgin Money story.”
But Unite union national officer, Rob MacGregor, responded: “Thousands of banking employees have this morning heard through the media that their jobs are no longer secure.
“Unite the union represents staff across both banks and has this morning expressed deep unease about jobs and services across both these financial institutions.
“The purchase of Virgin Monday by Clydesdale and Yorkshire Bank will change the face of banking in many high streets across the country.
“It is vital that the skilled and experienced workforce are given assurances that branches and contact centres will not be closed leaving customers without their much valued access to local banking.”
Shares in Virgin Money closed 2% lower and CYBG was down less than 1%.
Russ Mould, investment director at AJ Bell, cautioned it was far from a done deal amid jitters in the sector caused by TSB’s catastrophic IT migration failure.
He said: “While the target already has the backing of Virgin Money’s largest shareholder, Virgin Holdings with a c.35% stake, other shareholders will still have to approve the transaction.
“There is some strategic logic in putting them together but widely different cultures and the combination of two IT platforms are two major risks which could cause a hiccup or two down the line.”