To cancel an American Express credit card, you need to call them directly using the phone number on the back of your card.
Canceling an American Express card literally, can take less than 5 minutes (just call the number on the back of your Amex credit card, and speak to a representative). It may be one of the easiest calls you will make, and American Express customer service should not hassle you about canceling the card.
American Express states: "To cancel your Card, please call the number on the back of your Card or 1-800-528-4800 (24 hours / 7 days). If you are outside of the U.S., you may call collect at 1-336-393-1111."