A few questions:
What state are you in, and were you living in the same state when the supposed debt was created? If not, what are both states?
Was any attempt made to collect on the original debt, and if so, when was the last time an attempt was made?
Based on your response to my questions:
1999 was the last time they tried to collect this debt from you? That's not 10 years. It's almost 20. The statute of limitations in New York on credit card debt is 6 years. That doesn't mean some collection company can't buy an old debt and attempt to restart the clock on it - which is why you don't respond to them in any other way than to say 'PROVE IT!' - but I simply don't believe that American Express itself is attempting to collect a debt it hasn't attempted to collect in 19 years.
If the letter is actually FROM American Express, and not from some scammer using the name of the company you originally owed, then what you do is write back to that address with:
"Re: Your letter dated x, about a debt you say I owe. I deny that I owe this debt. You must provide proof that I owe it, if you care to continue wasting your time. The Statute of Limitations on credit card debt in New York State is 6 years. See New York Consolidated Laws, Civil Practice Law and Rules - CVP § 213.
This will be my only and final communication with you regarding this manner. Any additional attempts, by you or your agents, will be considered harassment and I will seek legal remedy as allowed by law.'
Send it by certified mail with proof of delivery and keep a copy for your records. The seven or eight bucks is worth every penny.
If the letter is NOT from American Express directly, send them the same letter but leave out this line: You must provide proof that I owe it, if you care to continue wasting your time.
If you do hear from them again, contact a lawyer to go after them for harassment.
Do I think it's right to not pay debts you owe? No, I don't. But the law is clear on what steps the creditor must take to collect/preserve their right to collect. If they don't take those steps, they lose and you win.