Amazon offers a simple way to turn the balance on prepaid credit cards (including gift cards) into Amazon store credit: just go to their "Reload Your Balance" page and enter your prepaid card as a payment method. This works perfectly for Visa- and MasterCard-branded gift cards. But there's a weird behavior with AmEx cards.
The way Amazon reloads your balance is kind of funky: they make a test charge of $1, which places a temporary hold on those funds. Then they try to bill the card for the amount of money you requested. But because there's a hold on one dollar of it, the charge won't go through, due to insufficient funds.
So, if you want to use a prepaid AmEx card to reload your Amazon balance, you need to add (at most) $1 less than the balance on the card.
If you've ended up at this page, you've probably already tried and failed to reload your balance with an American Express card at least once, so it's important to note that these $1 holds are cumulative. If you have a $100 gift card and have tried to add $100 in credit, the $1 hold is still in place, so your available balance is now $99. If you tried it twice, it's now $98, and so on. So you either need to wait a few business days for the holds to expire (so that you can purchase $99 in credit), or subtract $1 for each time you have tried and failed.
What are you supposed to do with a $1 balance on your prepaid American Express card? Amazon doesn't (as of this writing) allow you to split payment methods at checkout, so if you want to use the full balance at Amazon, you need to leave precisely enough money on the card to buy an item you want; buy a gift card for the value of the remaining balance; wait for the $1 hold to clear; and then buy the original item.
I'm told that some retailers' self-checkout lanes allow you to split payment methods (Walmart is apparently one) so you could use that $1 at another retailer. But unless you do some mathematical jiujitsu, you're going to be leaving a dollar on the table if you try to reload your Amazon gift card balance with a prepaid American Express card.