Just found another solution; the connector-plug of the ipod 1st and 2nd generation battery fits into the connector on the hard drive thermal sensor connector in the mac mini A1176. Find an old battery for the old ipod, cut the plug off with the wires as long as possible, connect those wires with the torn-off cable, and that's it. 50% chance you get it right first time; switch cables if the fan keeps going full blast.
Maybe someone can say what the plug is, so you don't have to look for an old ipod battery....