How to fix a leaking Grohe Adagio 37945 SH0 cistern?

I have a leaking cistern with a constant trickle of water into the toilet bowl through the overflow. I think that this is caused by the water inlet not being closed by the ball cock. I don't see any obvious means of adjusting the ball cock nor can I see or feel a stop valve to stop the water supply to the cistern.

The cistern is a Grohe Adagio 37945 SHO model.