Just standard surgery. I work at a VA hospital and we are whacking things off all the time. If we are dealing with magic, I supposed you could give them some supernatural form of gangrene. If that were the case things could drop off without surgery or risking shock, it would just take a while and smell really nasty. That's a smell you never forget, let me tell you.
I can't think of a way a non-medically trained person could do something like that without magic. You might get away with fingers and toes and maybe a hand or foot, but chances are the victim would die of shock pretty early on.