Considering the Casimir effect due to phononic excitations of a weakly interacting dilute BEC, we derive a renormalized expression for the zero temperature Casimir energy E_c of a BEC confined to a parallel plate geometry with periodic boundary conditions. Our expression is formally equivalent to a bosonic field at finite temperature, with a nontrivial density of modes that we compute analytically. As a function of the interaction strength, E_c smoothly describes the transition from the weakly interacting Bogoliubov regime to the noninteracting ideal BEC. For the weakly interacting case, E_c reduces to leading order to the Casimir energy due to zero-point fluctuations of massless phonon modes. In the limit of an ideal Bose gas, our result correctly describes the Casimir energy going to zero.