Diamond is the only known high-pressure structure of carbon. In spite of its fundamental and planetary importance, the stability domain of this strong covalent material is largely unknown. After decades of experimental efforts, evidence was obtained that the diamond-liquid melting line has a positive slope above the graphite-diamond-liquid triple point. At higher pressure, theoretical studies have suggested that the melting curve of diamond should have a maximum, owing to a loss of stability of the sp3 hybridization in the fluid phase. Accurate Hugoniot data of diamond exist up to 590 GPa (ref. 6). Higher-pressure measurements along the diamond Hugoniot have recently been achieved by laser shocks, showing that diamond probably melts to a conducting fluid. We report here laser-shock Hugoniot data across the melting transition. The shocked diamond crystal begins to melt around 750 GPa. Furthermore, a negative volume discontinuity at melting is observed. This requires a negative melting slope and thus supports the existence of a maximum on the diamond melting curve. These melting data allow us to test various calculations of the phase diagram of carbon at very high pressure. Finally, the stability domain of the diamond crystal is now constrained in a relevant region for Uranus-like planetary interiors.