1: pull off the back-plate and see if something is wrong with the tremolo springs. I had a similar problem (not quite this severe) so I added a spring, and although it stiffened up the action it solved the problem.
2: While you're behind the back plate, the tremolo springs are all connected to a piece of metal that is held in place by two long screws; if you tighten the screws further into the guitar body this can tighten down the tremolo by stretching the springs a little more. These are at least worth checking on before you go to more extreme measures.
