Superionic (SI) water ices—high-temperature, high-pressure phases of water in which oxygen ions occupy a regular crystal lattice whereas the protons flow in a liquid-like manner—have attracted a growing amount of attention over the past few years, in particular due to their possible role in the magnetic anomalies of the ice giants Neptune and Uranus. In this paper, we consider the calculation of the free energies of such phases, exploring hybrid reference systems consisting of a combination of an Einstein solid for the oxygen ions occupying a crystal lattice and a Uhlenbeck-Ford potential for the protonic fluid that avoids irregularities associated with possible particle overlaps. Applying this approach to a recent neural-network potential-energy landscape for SI water ice, we compute Gibbs free energies as a function of temperature for the SI fcc and liquid phases to determine the melting temperature Tm at 340 GPa. The results are consistent with previous estimates and indicate that the entropy difference between both phases is comparatively small, in particular due to the large amplitude of vibration of the oxygen ions in the fcc phase at the melting temperature.