While going over some records, we found an order from six months ago
that shipped twice.  It's worth about $1000.

We have copies of the FedEx delivery signatures, proving that the
company received both shipments.

We contacted them and asked if they would prefer to return the
merchandise (at our freight expense) or pay for them.

They responded that they don't have any record of placing two orders,
so they refuse to do either, despite proof that two orders were
delivered to them.

Our system is well designed to prevent overshipping from being
possible (we figured out the loophole.)  We have never had to figure
out what happens in this situation before.  Since this situation was
"impossible" for us, our terms and conditions don't address it. 
(Opps.)  The company just placed the order with us -- there is no
purchase order on their side for terms and conditions.

My question is: If the customer decides to try to rip us off for about
$1000, is there anything we can do about it?  We have already secured
an authorization on his credit card, however are not billing it until
we find out what the legal situation is.

This is tricky.

Because sometimes delivery notes are signed by people without authority.

Who knows?

Maybe the legitimate customer rejected the delivery because it was a
dupe and someone else in the chain seized the opportunity to work a
fast one.

Also, your having taken 6 months to discover the duplicated delivery doesn't help.

If I were you, I'd write it off to experience.

Look on the bright side: it's probably worth $1,000 for you to have
discovered the loophole.

did you try sending them copies of the two fedexes - it is in itself
proof and refutes their argument.  you could just invoice them and
handle it as a collection issue, make them prove they didn't receive
it.  if they are a big customer you  may want to make a strictly
monetary decision to  write it off - although i personally abhor
dealing with unreasonable or unethical business associates.   the six
month delay is something of a mitigating factor - it is aggravating to
get a very old invoice.

you could get a lawyer to write them a letter if he will do it for you
 for free or very little.  some people use small claims forms without
a lawyer although i have never done that.

one final thought,  they may think that they paid twice and           
       don't wish to or unable to track their payments back six
months.  that was my first thought when i recently got a very old bill
12 months i think from Xer--.

ps i am not a lawyer, just giving you some thoughts

Another thing that has occurred to me is that credit card transactions
have to be processed within a limited time frame, otherwise the credit
card holder can reject them anyway.

This is (or used to be) the practice in the UK.

Worth checking!