If you're going to go the heat shrink tubing route, you want some that has heat-activated adhesive on the inside to make the seal, not just 'regular' heat shrink tubing, as it will leak much more easily. At least a few companies make a waterproof tube with compression seals at each end filled with goop. With those, you can use wire nuts or clamps to make your connection, then this fits over it. They are not inexpensive, but are engineered to work. You'll probably want to cut out the corroded part, and unless you have some slack, you'll be splicing in a new section, so you'll need at least two of them, and maybe a third for the neutral and maybe even a forth for the ground wire. You can probably find what you need at a local electrical supply shop, but here's one that should work
http://www.walmart.com/ip/IDEAL-46-...316092&tmode=0000&veh=cse&sdc_id=964849408534