Assuming you're not going super-bling with this conversion, and assuming the bottom bracket cups aren't destroyed, just leave the fixed cup (the one that's being a pain to remove) in place. Figure out what crank you want to use, and bring everything to a well-stocked bike shop to find a spindle that (a) fits your bottom bracket shell and (b) gives you the right chainline with your crank.

You could try to obtain a modern, sealed bottom bracket, which has the cups, bearings, and spindle all built into a single unit. But because of the threading headaches mentioned in previous posts, I would just work with what you've got, switch up the spindle, and avoid the extra hassle (and potentially very high cost) of replacing the whole thing.