In the phase diagram of water, superionic ices with highly mobile protons within the stable oxygen sublattice have been predicted at high pressures. However, the existence of superionic ices and the location of the melting line have been challenging to determine from both theory and experiments, yielding contradictory results depending on the employed techniques and the interpretation of the data. Here we report high-pressure and high-temperature synchrotron X-ray diffraction and optical spectroscopy measurements of water in a laser-heated diamond anvil cell and reveal first-order phase transitions to ices with body-centred and face-centred cubic oxygen lattices. Based on the distinct density, increased optical conductivity and the greatly decreased fusion enthalpies, we assign these observed structures to the theoretically predicted superionic ice phases. Our measurements determine the pressure–temperature stability fields of superionic ice phases and the melting line, suggesting the presence of face-centred cubic superionic ice in water-rich giant planets, such as Neptune and Uranus. The melting line determined here is at higher temperatures than previously determined in static compression experiments, but it is in agreement with theoretical calculations and data from shock-wave experiments.