The thermal conductivity of ideal short-period superlattices is computed using harmonic and anharmonic force constants derived from density-functional perturbation theory and by solving the Boltzmann transport equation in the single-mode relaxation time approximation, using silicon-germanium as a paradigmatic case. We show that in the limit of small superlattice period the computed thermal conductivity of the superlattice can exceed that of both the constituent materials. This is found to be due to a dramatic reduction in the scattering of acoustic phonons by optical phonons, leading to very long phonon lifetimes. By variation of the mass mismatch between the constituent materials in the superlattice, it is found that this enhancement in thermal conductivity can be engineered, providing avenues to achieve high thermal conductivities in nanostructured materials.