The temperature dependence of the Kapitza resistance (thermal boundary resistance) at the interface between two dissimilar solids is calculated for a model system consisting of two semi-infinite, harmonic fcc lattices in register. The spectral density of phonon flux transmitted across the interface is obtained from a numerical calculation of the phonon transmission coefficient and group velocity and is used to calculate the Kapitza resistance. Results for the spectral density of the transmitted phonon flux, the spectral dependence of the phonon transmission coefficient, and the temperature dependence of the thermal boundary resistance are presented. Calculation of the thermal boundary resistance for real systems using the results of this calculation is discussed.