The melting curve of MgO is extended up to 4 TPa, corresponding to the Jovian core pressure, based on the one-step thermodynamic integration method implemented on ab initio molecular dynamics. The calculated melting temperatures are 3100 and 16 000 K at 0 and 500 GPa, respectively, which are consistent with previous experimental results, and 20 600 K at 3900 GPa, which is inconsistent with a recent experimental extrapolation, which implies the molten Jovian core. A quite small Clapeyron slope ([Formula: see text]) of [Formula: see text] is found at 3900 GPa due to comparable densities of the liquid and B2 phases under extreme compression. The Mg-O coordination number in the liquid phase is saturated at around 7.5 above 1 TPa and remains smaller than that in the B2 phase (8) even at 4 TPa, suggesting no density crossover between liquid and crystal and thus no further denser crystalline phases. Dynamical properties (atomic diffusivity and viscosity) are also investigated along the melting curve to understand these behaviors in greater detail.