Deuteron nuclear magnetic resonance spin-lattice relaxation times T-1 have been measured for deuterated benzene molecules (C6D6) in dilute solutions of water (H2O) and methanol at temperatures of 1, 2, 5, 15, and 30 degrees C and pressures up to 300 MPa. At the higher temperatures of 15 and 30 degrees C, the rotational correlation time tau(2R) Of C6D6 in the aqueous solution (H2O) increases with pressure, whereas the tau(2R) Of D2O decreases. This difference is interpreted in terms of the compression mechanism involving the packing effect and the orientational dependence of intermolecular interactions. At lower temperatures, the pressure dependence of the tau(2R) Of C6D6 at initial compression is smaller. Especially, at the lowest temperature of 1 degrees C, the tau(2R) Of C6D6 in the aqueous solution is very weakly dependent on the pressure up to 100 MPa. This implies that a strong hydration shell due to the hydrophobic effect resists compression up to 100 MPa and that the rotational mobility of C6D6 is not affected by the pressure. When the pressure is raised beyond 100 MPa, the pressure dependence of the tau(2R) at 1 degrees C is Similar to those at the higher temperatures of 15 and 30 degrees C and the hydration shell is relatively weak. For the methanol solution, in contrast, the tau(2R) Of C6D6 increases monotonically with pressure both at 1 and 30 degrees C. The activation energies of the tau(2R) Of C6D6 in the aqueous solution and D2O in pure liquid exhibit anomalous reductions as functions of the pressure. The reduction of the activation energy is accounted for by the pressure-induced weakening of hydrogen bonds between water molecules around the solute benzene.