The phase properties of water confined in mesoporous silica MCM-41 were investigated over a temperature range of 100-298 K as a function of pore size by specific heat capacity and inelastic neutron scattering (INS) measurements. The water content of the samples was carefully controlled to ensure the capillary filled state and no overloading of water. The values of heat capacity of the pore water are higher than those of bulk ice and liquid water over the whole temperature range measured. The contribution of water in the inner part of pores (abbreviated as the internal water) was elucidated by using the heat capacity data of monolayer water measured. The entropy of the internal water was then estimated from integration of the heat capacity of the internal water. The entropy values of the internal water increase by confinement in the pores of MCM-41 in both liquid and frozen regions, indicating an increase in the deformation of the structure and/or a change in the dynamics in both regions. The INS spectra show the density of states for the librational motion of water frozen at 50 K, suggesting that the confined water is similar to amorphous ice rather than to crystalline ice. When the sample is warmed to melt, the band edge of the librational motion for water frozen in large pores (diameter of 3.6 nm) shifts to a lower energy side, indicating the weakening of intermolecular hydrogen bonds. For water in small pores (2.1 nm), on the contrary, the librational band shifts slightly to a higher energy side, suggesting the low density liquid to high density liquid transition (L-L transition) at 225-250 K. A plausible mechanism of the L-L transition of water in confinement is proposed in terms of incomplete growth of homogeneous nucleation of ice due to an interfacial free energy effect to inhibit crystallization of water confined in small pores.