I emailed them just to make sure when I first got my card, and this was the answer I got:
Please be advised that our system only allows Cardmembers enrolled in Pay by Computer/Pay by Phone to pay more than their current balance due if it has been more than 30 days since their enrollment or last banking information change. Since you enrolled in Pay by Computer/Pay by Phone on 07/28/08, this feature will be available to you once the 30-day period has expired.
At that time, you will see a new option available when you log in to pay your bill online. This will allow you to pay your total outstanding balance including new charges accrued since the last statement.
Alternatively, you may send your payment to us at the address on your statement, or to our main remittance center listed below.
REMITTANCE ADDRESS FOR U.S. MAIL:
American Express
PO Box 360001
Ft. Lauderdale, FL 33336-0001
OVERNIGHT PAYMENT ADDRESS (Federal Express, etc.):
American Express
US Payment Florida
2965 West Corporate Lakes Blvd
Weston FL 33331-3626