In this paper we prove the local exact controllability to the trajectories of the Cahn-Hilliard equation, which is a nonlinear fourth-order parabolic equation, by means of a control supported on an interior open interval. To prove this result we derive a Carleman estimate that allows us to conclude, thanks to a duality argument, the null controllability of the linearized equation around a given solution. Then, we apply a local inversion theorem to extend the control result to the nonlinear equation.