Now building societies are following suit. The logic is irresistible: the branch of the future is a home computer, or even a television set, not a building.
Earlier this month the Co-operative Bank became the third bank to launch an internet banking service, which can do almost anything a branch can do, and more than a cash dispenser. Anyone with a Co-op personal account and an internet link can use a computer to carry out most day-to-day banking business - paying bills, transferring funds and direct debits.
Nationwide and Royal Bank of Scotland launched internet banking services last year. The Co-op's move is more evidence that banks see the internet as by far the best alternative to the branch.
Datamonitor, a research group, predicts that 9 million Europeans will use internet banking within three years.
The Co-op believes its internet banking service is the most comprehensive on offer. The bank says around two-thirds of transactions can be done over the internet, with the rest relying on branches or phone banking.
The Co-op's product is significant, too, because of its technology. The bank developed the site in Java, a software technology that allows it to produce pages that will run on any computer with a recent internet browser, regardless of the make of PC or operating system. At the moment, Nationwide and RBS only support computers running Microsoft Windows.
The Co-op site also works with Apple Macintosh machines, and the bank has prototypes for a Hewlett-Packard hand-held computer. Java also makes it easy for the bank to update its systems to work with new technologies as they emerge, including digital television.
The pages of the Co-op's easy-to-use site have a designer rather than a banking feel. "We hope it is intuitive and the design is friendly," says Keith Girling, head of channel development. "We felt that a lot of internet banking services are boring, and look like banking services."
Mervyn Pedelty, chief executive of Co-operative Bank, says: "In future, as all banks share the same channels, customers will see banking as a commodity, even more than they do today. Banks will need strong brands and quality products. Historically, banks have been woefully inadequate in customer services."
Mr Pedelty's view is that banking will become an icon on a computer monitor, and deciding where to bank will be as easy as clicking a mouse. "It could mean the end of traditional banking. High-street banking is declining pretty fast these days."
For now, potential is limited by the public's confidence in the internet and the number of people who have computers at home. "We believe one in four of our customers are at ease with the technology," says Mr Pedelty. "Others may join the group, but it depends on the success of the internet as a whole."
He points to the take-up of telephone banking to support this view. "Since the early 1990s telephone banking has grown enormously; that shows how quickly consumer habits can change. Potentially, the net can have a far greater impact."
Security is one reason customers may hold back. The risk of fraud via electronic banking is very small, but the internet still has a public image for lawlessness. The Co-op, like its rivals, has invested heavily in electronic security measures and, as an added reassurance, it asks customers to register for on-line banking by phone.
The threat to bank branches is another reason not everyone is comfortable with the idea of internet banking. Banks are being criticised for closing branches in rural and inner-city areas.
But the Co-op's relatively small branch network means electronic banking gives the bank a chance to expand into new markets. It is working on a touch-screen computer to let people use internet banking in its unstaffed kiosks.
"The revolution is not about any one technology," says Mr Pedelty. "It is about diversity, but more importantly, customer choice." Co-op has a link with satellite television and a deal with Vodafone that lets customers check balances from a mobile.
Electronic banking has the potential for more than convenience. Banks can give real-time information about accounts, and it should be possible to shorten the three-day delay when money moves between banks. Internet banking saves money, too, once set-up costs are covered. A bank that can persuade a large number of customers not to use branches will see profits rise.
The danger is that people without the financial or technical resources to use the internet will be left behind. This already happens in the United States, where bank charges come in two tiers: one for people who use branches, and a lower one for those that do not. Technophobes and the less well-off lose out.
The Co-op's solution is to provide the computers itself. It is setting up "cybercafes" with internet-linked computers in six of its largest branches, and has plans to put touch-screen terminals in public places such as supermarkets. In a few years, as cost fall, it might even pay banks to subsidise or even give away a basic computer to customers willing to make the switch.
Co-operative Bank web site: http:///co-operativebank.co.uk.