How do I move a flash website to a different server when the Flash URL's are hard-coded

If you are moving the domain to a different server as well then you don't need to change anything.

If however, you mean that you are hosting the swf's under a different domain you will have to alter the hard coded URL's within the FLA's.

As you don't have them to hand you can try using a decompiler to create FLA's from the swf's.

http://www.sothink.com/product/flashdecompiler/index.htm

I believe there is a free trial version you can have a go with.

I have just downloaded the SoThink product trial.
I will attempt modify and compile the FLA file...

The SoThink free trial version only decodes the first 2 action script files.
I have tried the free Flare action script decompiler, but had no luck in recompiling in flash.
Is it possible to catch the output of the swf file on the server and change it?

No, the URL link is actioned in the users browser and therefore goes straight to the domain IP address according to the particular DNS that is being used.

Editing the flash file is the only way to get around this unfortunately.

I finally got management's permission to purchase SoThink decompiler.
Had site up and running in less than 2 hours.
Thanks.

First answer was partially right because it alluded that the free version might work.  It didn't and I had to buy it in the end.  I was trying to be nice that you gave me advice that I was already aware of and I awarded all of the points to you.