Depends on the insulation. Often, the best way to handle this is with a solder pot. Just dip the end in, it will melt the insulation and tin the end. (This will not work with Teflon but it's worth a try otherwise.)
thanks Kal, after posting this thread I also found a procedure written on vinylnirvana.com that used the same Cardas wire I'm using - said to burn it off using a butane lighter. This should be much better than what I was trying. I like your suggestion better because it helps solve my other problem (ie working with the tiny fraying wire) by tinning the end.
Is this the Litz wire? I re-wired an arm once with that and found that I got pretty good results by burning the insulation off (40 watt iron) and then applying a generous amount of flux to the end of the wire before tinning. I also used an alligator clip on the wire as a heat sink to help prevent the heat from traveling.