Duplex

[BrandtCooley]

There is Brick, Sheetrock, Brick seperating me and my neighbors, which would be fine if there wasn't a door that that doesn't block even a cellphone conversation.  I'm planning on doing some recording (the loudest thing is going to be a somewhat cranked guitar amp) and I want to soundproof that door.  The door doesn't need to be opened and it's got a recess of about a foot that I can stuff with whatever I want, though I don't have much money.

[johnbergstromslc]

First of all, you want to seal up the door.  If, as you said, the door doesn't need to be opened, get some silicone caulk and seal up the perimeter.  Then put some kind of sweeper strip along the bottom (I once used just a board, screwed to the bottom of the door) and seal up the bottom too.  Any little crack or gap can transmit sound energy.  
Then, fill the cavity full of fiberglass insulation and put a couple sheets of drywall on your side of the door jamb.  Seal those up too.  
Fiberglass is pretty cheap ($10 for 40 sq. ft.), drywall is about $7 a sheet, and the caulk is $3-4 a tube.