Monte Carlo simulations show that a system of hard, spherical particles with a short-ranged attractive interaction, can undergo a first-order transition from a dense to a more expanded solid phase with the same structure. This phase transition is analogous to the liquid-vapor transition in systems with longer-ranged attractive forces. In particular, the solid-solid transition terminates in a critical point. We argue that this phase transition should be experimentally observable in certain types of colloidal crystals.