The melting curves of ice VI and ice VII were measured using a diamond-anvil pressure cell up to P=2.8 GPa and T=400 K and their metastable extensions. We found the existence of the metastable extension of the phase boundary between ice VI and liquid water within the stable region of ice VII and confirmed the presence of the metastable extension of the phase boundary between ice VII and liquid water within the stable field of ice VI. We found that the curvature of the ice VI-ice VII phase boundary is more pronounced than previously reported. The complete interpretation of our results might be made on the basis of thermodynamics of liquids in small confining systems.