Forgive my inaccuracy.
Say you have an antisocial person who stole a wallet. Off with the head.
Ouch.
What purpose does the death penalty serve? To get those who are dysfunctional out of society, because banishment isn't going to work. If you are a person who cannot function inside of the law, who will ignore things for your own gain, you do not belong in society.
Your stance raises the interesting metric of "how one knows when rehabilitation isn't possible," which to me seems most likely to be "attempt to rehabilitate a person multiple times until it becomes clear it is ineffective." This puts aside at least some of the immediacy of the worry of cutting someone's head off simply because he has stolen a wallet, AnSionnachRua - although I don't claim it's a perfect suggestion.