Any vacuum leaks in the system will result in a firm'er brake pedel....NOT one that is soft and goes to the floor. You mention that the engine is shutting of, which is also a good sign that you're not dealing with a vacuum leak.

It really sounds like the seal in the master cylinder is shot. When you press the brake pedal all the way to the floor, either that brake fluid has to be pushed SOMEWHERE or if it's not, then it's staying in the master cylinder because it can't be pushed out (which is the job of the master cylinder, right?)

Granted, you can also get this sort of behavior if you have air in the lines too. You press the brakes and the fluid gets pushed out but then it just starts compressing the air that's in the lines....allowing the pedal to be pushed more.

Have you had any brake work done recently?