I used to do this before leaving for summer recess in college.
Paper statements come with a 'payment coupon'. The front of the stub says 'please enter account number on all checks and correspondence' and has:
- account #
- billing address
- Amex mailing address (depends on location)
East Coast:
American Express
P.O. Box 1270
Newark, NJ 07101-1270
West Coast:
American Express
P.O. Box 0001
Los Angeles, CA 90096-0001
I don't have the addresses for the other regions. However, I once sent a check to a location which was not my assigned one, and the payment went through just as promptly.
I have been paying online ever since the service was offered. Not one mishap.