After the hacker has his address and copies the addresses he has sent emails to in the past (stored in his email program), mail can be sent without using his PC at all.

There are sites like

http://anonymailer.net/

that have made this easy to do. There's not a lot the owner of the stolen address can do about it. If you explore the header of a fake email, you may be able to find out which service originated the message and send them a complaint.