The ATF rejected my form 1 for a MKE AP5-P, stating the serial number is a duplicate of an already registered firearm. I double checked the serial number on the form prior to submission, and checked it again after the rejection and the serial submitted is what is engraved on the firearm.

How do I fix this problem?

That's a fun one.  You used the full serial number right, Including the "T0624" part?  And is this a new gun or did you buy it from someone else who might have already Form 1'd it in the past?

You'll have to try and call their help line or use ask the experts.  Not sure how far they will go but I'd guess the issue is someone else fat fingered their form and put the wrong number.  Obviously they won't share with you who already used your number, but maybe they'd be willing to contact that person to try and verify their number is correct.

Somebody else fatfingered theirs and got a stamp for the wrong serial number and have been walking around with an unregistered SBR.
RIP to their puppy.

Well. I made a mistake. The serial number is on the top of the receiver, just forward of the rear sight tower. I incorrectly assumed that the numbers engraved under the forearm constituted the serial number. I installed an optic rail which covered up the serial on the top, so I missed it.

They will refund your money and you begin again, no biggy nowadays.

Oops.

For future reference, the serial will always be on the receiver.  Numbers on the barrel or elsewhere may or may not be the serial number.  
As the poster above said, just resubmit and you should be good to go.

Resubmitted with actual serial number on Tuesday, approved this morning.