If the barrel is installed on a receiver, you have to pay the assembly/disassembly fee.
http://i264.photobucket.com/albums/i...psac9f2085.jpg
For your reference, my AA barrel above is thicker than the standard Government profile. It is a straight .700" under the handguard, instead of tapered down to .600" (IIRC) for the Gov't profile.
If you send your barrels into ADCO and don't ask for a specific diameter, their site says "Unless a specific contour is noted in the comment section the barrel will receive our standard medium weight contour", but I couldn't find what that might be anywhere on the site.