Martin,
When the "kill" switch is flipped to shut the saw off you are essentially grounding the magneto to the saw frame. What has happened is that your "ground wire" has broke (typically) or has come out of the off switch. Take your air filter cover off. Operate the kill switch and see if you can see it moving. If you can't you may have to take off the air filter as well. You should see on the inside of the case opposite of part of the switch you operate with your finger a silver barrel part that is about 5/8" long with a black wire hooked to it. Since this part moves and the wire flexes here it typically breaks right at the end of the silver connector. The other end of this wire feeds through the case to the magneto and usually plugs in with a 1/4" flat terminal. If the wire is broke, if you can solder you can fix it. If you can't you will have to find a way to attach this wire back onto the silver end so when you flip the switch off that silver end make contact with a piece of thin spring steel that is screwed to the case so it grounds it out to kill it. If you can't then you will have to order one online or visit your local Stihl dealer and get a replacement and feed the wire through where the old one was and hook it back up.
Luggnutt, February 2012