We present melting temperatures determined at several pressures for solid argon, krypton, and xenon up to 47, 18, and 12 GPa respectively. At these pressures the observed melting temperatures rise to 2790±150 K for argon, 2175±150 K for krypton, and 2054±150 K for xenon. These data suggest that rare gas solids (RGS) melt at high temperatures relative to other classes of materials with increasing pressure. The argon melting temperatures are consistent with both empirical and theoretical predictions. Within experimental error, the melting curves for Ar and Kr are consistent with the Simon-Glatzel (SG) expression for the melting curve obtained at lower pressures. For xenon, the data indicate that melting occurs at lower temperatures than predicted by an SG relation based on melting data to 0.65 GPa, but above the melting curve expected from a corresponding-states scaling of the internally consistent melting curves of both Ne and Ar. In addition to the intersection of the argon and iron (Ar-Fe) melting curves in the range 42-47 GPa and 2750±150 K, the experiments suggest an intersection of the Kr-Fe melting curves near 23 GPa and 2300±150 K and of the Xe-Pt curves in the range 12-15 GPa and 2300±150 K.