Abbey National said the deal would help fulfil its aim of diversifying its profits stream away from its traditional mortgage and savings activities.
Cater Allen's chairman, James Barclay, said that although the offer had been welcomed and recommended by the board, "it is also sad news: sad that after 180 years the company is to lose its independence".
Abbey National is offering 580p for every ordinary Cater Allen share. This represents a 31 per cent premium to the closing price on 4 June, the day before Cater Allen announced it had received an approach that may lead to an offer.
Peter Birch, Abbey National's chief executive, said: "The acquisition offers a unique opportunity to increase Abbey National's strategically important growth market which builds upon our existing treasury and offshore business."
Abbey has been keen to diversify into these areas. Last year 40 per cent of its profits were earned from non-traditional businesses. The bank intends to increase this figure to 50 per cent by the end of the decade.
Abbey's treasury and offshore businesses last year contributed more than 20 per cent of the group's pre-tax profits.
"Abbey National's ambitions in these areas would be difficult to achieve organically," Mr Birch said.
Cater Allen reported its full-year results yesterday, showing profits of pounds 17.1m compared to pounds 14.6m the previous year. The dividend was held at 30p.
Cater has its origins as a discount house dealing with the Bank of England and is still active in its traditional money-market activities. But, in common with other houses, it has found it difficult to turn a profit in these areas and has sold its futures trading operations.
Instead it has diversified into offshore banking with profitable operations in Jersey and the Isle of Man. Abbey National has two similar operations in Gibraltar and Hong Kong. Another Cater business which may have attracted Abbey is its ownership of City Deal, a private client stockbroker. A few months ago it added Glasgow-based stockbroker Aitken Campbell.
Abbey National shares closed 1.5p lower at 628.5p.