The fact that it should be synced, but isn't, begs the question: what is actually wrong with your system?
What exact version of After Effects are you using (down to the decimal points)? You say AE6, but I'm assuming you mean CS6 (which is actually version 11 and you should be on version 126.96.36.199 until the update comes out later this month). If you're not on version 11.0.4, update! See if that helps.
What version of QuickTime are you running?
What codec is your MOV file encoded with?
What are your system specs?
OK yes, I am on AE CS6 version 188.8.131.52
QT version 10.3
Not sure on the codec, it says libx264 but I have no clue what that is. The file was provided by the client so I didn't have any control over that part.
Running on a Macbook Pro OSX 10.9.2, 500GB SSD, 2.3 GHz i7, 8GB RAM, NVIDIA GeForce GT 650M 512 MB
After seeing that codec on the file I am thinking that might be the issue, but I'll have to test it out.
I'd be curious if it rendered synced or not.
Also, does Premiere read it correctly?
If it plays in sync in Premiere, save yourslf some grief: export the file as a quicktime movie in the PNG codec, or ProRes 422 if you have it. Yeah, it's a pain, but when it gets into AE, it'll be right.
Treating the symptom only:
Duplicate the layer. Use the eyeball and audio switches in the timeline to turn video only on for one layer, and audio only for the other. Then slide the audio layer to the desired position.