What kind of wire are you using? A Nickle Cromium wire probably available from radioshack and a 12 volt battery should work quite well.
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While i haven't done anything like this, i think that a band saw would work pretty well. Just a guess though.
As far as your "wire saw" goes, be careful in regards to the fumes produced. I would make sure that you had plenty of ventilation and were wearing some type of mask.
In terms of what you are using for this wire saw, if you've got a good sized variac with a fuse on it, that's what you need. I would not use copper wire for several reasons. As Rives mentions, nickel chromium wire ( the stuff that they use for toasters ) would be ideal. I don't think that Rat Shack has this, but it would probably be worth checking out. Sean
The copper wire's resistance was probably too low for the battery. Use what Rives suggested.
Be careful with the variac. Most of them are not isolation tranformers, so your wire "blade" and its associated wiring will be electrically "hot". Touching it and a ground simultaneously would be very unsafe.