BIRMINGHAM, Ala. — AmSouth Bank is extending by two months its offer of free Internet banking and bill payment for life, thanks to surprisingly strong demand.

The $38.8 billion-asset bank signed up 160,000 new and current customers for online banking during the first three months of the offer, and it still is averaging more than 11,500 applications a week.

With the help of the special offer, AmSouth’s number of online customers has grown 380%, compared with the total in May 2000. It currently is averaging about 2,300 applications a day, compared with last year’s average of about 100 a day. April 30 was the busiest day so far, when more than 5,000 signed up.

AmSouth has notified current Internet banking customers that they need not pay for the online bill-paying service, which normally costs $5 per month.

In addition, for current customers and those who sign up by July 31, AmSouth is waiving fees for Banking with Microsoft Money, a service that integrates Microsoft’s financial management software with AmSouth’s Internet banking service. Banking with Microsoft Money normally costs $8 per month. Customers also do not have to pay for the software or setup fee, saving $9.95. In the first year, savings total more than $100.

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