You need to fill in the CAA documents and fax them to the CAA
You also need to fill in some FAA documents and fax them to the FAA.
You will then get a letter from the FAA confirming the application and tellling you to book an appointment with the FSDO to have an application interview.
You then turn up for the interview...fill in a load more paperwork (ensure you take your logbook and license). At this point the FSDO officer will decide if you can have an FAA license based on your JAA one. They might insist on a test flight to make sure you wont kill yourslef. If you have not passed the FAA pre solo you will have to do that exam. They might also insist that you have an FAA medical.
If they are happy to issue you with an FAA license, you will get a piece o paper stating that it is a temporary license and is valid for 120 days. At some time during the 120 days a nice green credit card style FAA PPL will arrive in the post.
I used the Albany FSDO...very very helpful and friendly. Plus i know a few of the bods there.