how can I fix the water that constantly runs through my toilet?

Although the ballcock on my toilet is up fully, water is still running through to the pan constantly. I've tried taking the whole mechanism apart to see if there was anything in there catching, but this hasn't worked.
I could be wrong, but it look like it counts on air to be flowing into the valve for the water to run - basically when the ballcock is up it should stop the air and subsequently the water. As I say I could be wrong.
Any advice/help is welcome. Also, if I do need a new part - if anyone knows where this sort of thing could be ordered then great!