19.5 HP B&S problem

Hi, I have a grass cutter that has a 19.5 HP B&S engine. This engine has developed an odd problem. I bought a battery for it last season, and it has been operating at its peak. All of a sudden, it is stone dead. A multimeter reads 0.00. So I decide to jump it. As I hooked up the connections, the engine started to turn over, and the key was in the off position.
I don't understand why this would be happening, The engine should not turn over when I jump the battery, unless something is wrong. Can anyone help to point me in a direction what would help to solve this problem?

Something is certainly wrong! You may have a defective key switch, or a defective starter solenoid. Let's try to narrow this down. On teh solenoid, there's one or two small connections in addition to the two large ones. Disconnect the small connection(s) and hook up the jumpers again. If it STILL tries to start, (with the key off) then I think your solenoid is stuck on). If that made a difference, I believe it's the key switch.

Many thanks, I will try that. I am very glad that I have some direction now!