In order to clarify the origin of the hysteresis between freezing and melting of pore water, we performed x-ray diffraction measurements of water confined inside the cylindrical pores of seven kinds of siliceous MCM-41 (a member of ordered mesoporous materials denoted by Mobil Oil researchers) with different pore radii (1.2-2.9 nm) and the interconnected pores of Vycor glass as a function of temperature. The hysteresis effect depends markedly on the size of the cylindrical pores: the hysteresis is negligibly small in smaller pores and becomes remarkable in larger pores. This strongly suggests that the hysteresis is arisen from size-dependent supercooling of water confined to the mesopores. For the water confined to the mesopores with pore radius of 1.2 nm, a continuous transition between a liquid and a solid precedes the first-order freezing transition of the pore water which would occur by the same mechanism as in bulk water.