Time-of-flight (TOF) inelastic neutron-scattering spectra were measured on beta-cerium (double hcp) and gamma-cerium (fcc) near the phase-transition temperature. Phonon densities of states (DOS) and crystal-field levels were extracted from the TOF spectra. A softening of the phonon DOS occurs in the transition from beta- to gamma-cerium, accounting for an increase in vibrational entropy of DeltaSvibgamma-beta=(0.09±0.05)kB/atom. The entropy calculated from the crystal-field levels and a fit to calorimetry data from the literature were significantly larger in beta-cerium than in gamma-cerium below room temperature, but the difference was found to be negligible at the experimental phase-transition temperature. A contribution to the specific heat from Kondo spin fluctuations was consistent with the quasielastic magnetic scattering, but the difference between phases was negligible. To be consistent with the latent heat of the beta-gamma transition, the increase in vibrational entropy at the phase transition may be accompanied by a decrease in electronic entropy not associated with the crystal-field splitting or spin fluctuations. At least three sources of entropy need to be considered for the beta-gamma transition in cerium.