You need an FAA license to fly an N-registered aircraft in the States. The usual way of doing this is to get a "license based on a foreign license", which is simply a matter of paperwork.
First of all, find the appropriate form on the FAA's website. (Someone will have a link to it, I don't have it handy.) Fill it in, and send it to the FAA. You will need to nominate a FSDO - your chosen school or club in the States will be able to tell you which is their nearest FSDO.
Then (and I'm assuming here that your JAR license is UK-issued) find another form on the CAA website, and fill this form in. Send it to the CAA, together which a cheque. They will cash the cheque, and then contact the FAA to inform them that you really do have a license. Once the FAA receive this confirmation from the CAA, they will write to you to tell you that your license can be collected from your nominated FSDO. Make sure you take this letter with you, go into the FSDO, pick up your license. Then, get a BFR from an FAA CFI, and go and fly!
You say you're going "in a few weeks time", though... that might be a little tight for time, depending what you mean by "a few".
FFF
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