To study the liquid-solid and solid-solid phase transitions of a confined O-2, we performed x-ray diffraction measurements of O-2 confined inside the cylindrical pores of six kinds of regular mesoporous adsorbents (MCM-41 and SBA-15) with different pore radii (R=2.1, 2.9, 3.9, 4.5, 5.3, and 7.0 nm) as a function of temperature. Hysteresis effects between the gamma -to-beta and beta -to-gamma solid-solid phase transitions are considerably larger than those between freezing and melting and the depression of the beta -to-gamma transition point is slightly smaller than that of the melting point. This suggests that the beta -to-gamma rather than the gamma -to-beta transition point represents the thermodynamic phase transition temperature between the gamma and beta solids. The nature of the phase transitions between the gamma and beta solids is almost unchanged by confinement. The depression of the melting point is analyzed based on a simple capillary theory by invoking an empirical relation between liquid-solid interfacial free energy and enthalpy of fusion proposed by Turnbull. The O-2 confined to the cylindrical mesopores of R=2.1 nm gives rise to an amorphous diffraction pattern even at the lowest temperature studied (27 K). Unusual behavior of the peak width against temperature strongly suggests that the amorphous pattern originates from the formation of the gamma -phase clusters rather than the vitrification of a confined liquid.