Sealing exterior junction box
I decided to fill the hole with something called "duct seal" from my hardware store.  It's like plumber's putty.  I stuffed it in the hole and molded it around the wire.  It seems to seal the hole up pretty well and is supposed to be weather-proof.

In retrospect, I probably should have bought a junction box that has a hole in the back and put it directly over where the wire comes out of the wall.  I noticed that some have caps that can seal unneeded holes but not the model I bought.