Step 1: The Problem
Step 2: Chop It
The short in my adapter was so close to the point where the cord enters the casing that I needed to open up the case to get to the length of good cord inside. Apple doesn't expect these cases to be opened and, as such, they super-glued them shut. This makes opening them a real pain.
Step 3: Get Inside
Once the seal is cracked, you still aren't quite home free because there's a metal shield inside that's glued to the top of the case. If you're pulling and it springs a little bit but doesn't open, get up under the shield with a screwdriver and gently separate it from the top. You can see the shield in image 2, below.
Step 4: Expose Some Wire
Start by carefully stripping off the outer casing. Wire strippers could work here, but be careful not to cut through the ground wires. I didn't have a wire stripper, so I used a knife and my fingernails. Once the outer casing is off, separate the ground wires and strip the casing from the inner cable. There will be some thin green floss mixed in to the ground cage that provides padding and strength to the cable that you can separate out and trim off.
Now we need to get a some exposed wire on the adapter side to splice the cord on to. See that little plastic thing that the cord passes through as it enters the case? It keeps the cord from getting yanked off of the board when you pull on it. Inside the plastic thing, the cord is tied in a knot. If you can cut away the plastic thing without cutting through the cable, you can untie the knot and get a decent length of wire to work with. As you can see in the second photo, I couldn't, but it worked out anyway.
Once you have the cord exposed, repeat the stripping process from earlier and splice away. To do this properly, you should probably twist the wires and then bind them with solder. I didn't feel like getting out my soldering iron so I just wrapped them with electrical tape.
Step 5: Wrap It Up
When you're putting things together, wrap the cable in such a way as to protect both the splice point and the board connection from stress. You don't want all of your hard work to be undone by a careless yank on your cord.
Once you've got everything pretty secured the way you want it, tape it up (or glue it, or encase the whole thing in carbonite).