I've used these with success:
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these work well on smaller wire when the entire run is fully supported and there is NO CHANCE of tension being applied to the run
you can use shrink tubing over them just for insurance
Nylon crimp and waterproof heat shrink over top. Just make sure to slide the heat shrink on before crimping.
You can also use the waterproof crimps or the solder ones but I trust crimping more than solder and to Steve's point soldering doesn't comply with AYBC standards. Guidelines like AYBC or building codes are there for good reason and it's best to follow when possible.
for larger wires and runs that may not be fully supported this method has more integrety
BUTT ... my personal preference for this situation would be to run a new wire
and there will be no question
a new piece of wire is cheap over the long haul ... specially since there will be no opportunities for hidden gremlins to attack anywhere in the middle/hard to access sections of the run