The bank said it had already achieved its targets for a five-year business plan, attracting pounds 5bn and 500,000 customers by offering market-beating rates of interest.
Mike Harris, the chief executive of Egg, said the bank would try to steal a march on the Internet banking market, aiming to attract 2 million customers by 2004.
Egg has attracted huge volumes of business by offering uneconomic interest rates above its rivals - currently 6 per cent on savings accounts. The rates entail heavy losses for its parent company, the Prudential.
Mr Harris said the need to stem losses - estimated to reach pounds 170m this year - was a factor in the decision.
"The decision we were faced with was - do we continue to invest in telephone commerce and build a new call centre at a cost of pounds 50m, when everything's moving to the Internet? In three years we think it will be as ubiquitous as the telephone," he said.
He added that the running cost of an Internet bank account is four-times lower than by telephone. Mortgages and personal loans will still be available from Egg by telephone.
The aggressive move is in sharp contrast to the strategy of Abbey National, the mortgage bank, which yesterday said it was turning its back on Internet banking.
Andrew Pople, head of retail banking at Abbey National, said: "We do not see [the Internet] as something that will dominate for the next two to three years. As a way of getting new customers I don't think the human being could be bettered."
Mr Pople described the Internet as an unpleasant buying experience inappropriate for financial products.
"The Internet is long-winded to get through, the sites are over-complicated and the choices are bewildering," he said.