Spigot/spicket replacement in tight quarters
I wanted to post what I did to fix the problem in case it helps anyone. I did not have to do anything with the drywall or damage the outside of the house either. The most difficult thing to do was getting the solder into the wall to apply to the joint.

I purchased the following items from Home Depot.


Plumbing soldering kit that came with propane torch, solder, flux.
Copper 1/2" male adapter so that I could simply screw the new spigot on: http://www.homedepot.com/p/1-2-in-Copper-C-x-MPT-Adapter-C604HD12/204620255
Threaded 1/2" spigot: http://www.homedepot.com/p/1-2-in-Sillcock-Multi-Turn-Female-Thread-to-Pipe-Valve-UV65204/206827980
Thread tape


I fixed it as follows: 


I tore up an old sock and stuffed it in the copper pipe to soak up the water.
I used some sand paper to clean off the old solder on the pipe.
Remove the sock.
Apply flux paste to the outside of the copper pipe coming out of the house.
Push the male adapter onto the copper pipe. Make sure the flux paste was applied to the correct length.
I had a very tight spot to work with, and the solder wire needed to reach back into the wall to where the joint was at, so I went ahead and straightened out a piece of solder about 7 inches long with a tiny bend at the very end. This is the only way I could apply the solder without removing material from the wall.
I used a water bottle to spray a good bit of water into the wall to dampen everything up.
Fire up the torch and apply heat to the male adapter. I couldn't apply heat directly to the joint since it was inside the wall and I didn't want to catch anything on fire. It took a while to heat the entire adapter up.
After the male adapter is sufficiently heated up, I poked the solder into the wall and applied it to the joint. I then pulled it back out, made another tiny bend at the end of it, and stuck it back in on the other side to make sure that it was soldered all the way around.
After letting it cool off, I applied the thread tape to the male adapter and screwed on the spigot. I turned on the water and made sure it works.


It seems to be fixed. It's been a few days and I haven't noticed any leaks.