I've tried splicing some wiring
of that guage in my VHF
microphone with very poor results. Fortunately, the wife of a friend does electronic assembly and she was able to do a superb job. Personally, I'd either take it to an electronics
repair center and have them solder each into a connector (preferably watertight) and then plug
them together on the new boat. You might be able to do the job yourself if you have any skill in soldering things like 22 ga wire, but I would still suggest soldering each piece to a connector rather than directly.
I suppose you've already considered just buying
a new cable? Pricey I'll bet.