A £14 billion loan funded by the Bank of England will allow B&B's retail deposits to be transferred to Abbey, with a further £4 billion to be paid by the Treasury to cover those deposits not protected by the scheme.
The Treasury said it will be "business as usual" for savers and borrowers at B&B. Branches will remain open as usual with call centres and internet banking also available.
The deal will mean that Santander will have an overall 10% share of the retail savings market with 1,286 branches from the combined business of Abbey, Alliance & Leicester and B&B.
B&B's shares have been cancelled, with compensation to be paid in "due course", the Government said.
Kevin Mountford, head of banking at moneysupermarket.com, said: "After weekend of speculation, confirmation that Bradford & Bingley is to be broken up will come as a surprise to nobody. What is a surprise is that Santander seems to have walked away with such a bargain. In the new world where retail deposits are king Santander has landed all £21bn deposited by B&B's savers at a price that looks extremely cheap. Whilst many of those savers will no doubt feel relieved this morning, we will need to wait and see whether the remains of B&B keeps its separate FSCS registration, or whether B&B savers who also have money in either A&L and Abbey will need to move it again to stay below the £35,000 compensation limit.
"The big question for UK consumers this morning, is whether B&B is the last domino to topple, or whether more banks and building societies will go the way of HBOS, Northern Rock, A&L, B&B, Cheshire and Derbyshire and fall victim to the credit crunch."
Louise Cuming, head of mortgages at moneysupermarket.com said: "For people with mortgages with Bradford & Bingley this is a worrying time. With the mortgage book now, effectively, being run down by the Government there is no incentive for them to keep rates competitive. The nightmare scenario for borrowers is to find themselves stuck with an uncompetitive mortgage but be unable to remortgage to a better deal elsewhere because they don't meet the lending criteria."