There's often a float-operated valve to control the water flow into the cistern. As the float rises on a lever the lever arm acts on the inlet valve and shuts it off.
You can usually fix the problem you have with a simple adjustment but sometimes the seal in the valve has perished and will need replacement.
Flush the cistern a few times and watch the float drop and rise up with incoming water. Look at the lever and see how it closes off the valve. Pull up on the lever arm and turn off the water flow by hand...
You need less water so either lower the float into the water or if there's a little nylon adjusting screw turn it in towards the valve and tighten the locking nut.
Good luck...
Peccavi
September 2009