We have made a high-resolution quasielastic incoherent neutron scattering (QENS) study of the translational dynamics of supercooled water contained in micropores of Vycor glass at different hydration levels. QENS spectra from the confined H{sub 2}O are analyzed in terms of the {alpha}-relaxation dynamics predicted by mode-coupling theory of supercooled liquids and by a recent computer molecular-dynamics simulation of extended simple point charge model water. We verify that the stretched exponential relaxation description of the long-time test-particle dynamics is consistent with the measured QENS spectral line shape. We are thus able to determine the wave-number dependence of magnitudes of the structural relaxation rate 1/{tau} and the stretch exponent {beta} as functions of temperature and coverage. A power-law dependence of the average relaxation time on the magnitude of the scattering vector {ital Q} is observed. In the {ital Q} range studied, the exponent starts out with nearly {minus}2.0, at room temperature, indicating a continuous diffusion, and gradually becomes less negative as the temperature is decreased to below the freezing temperature. thinsp {copyright} {ital 1999} {ital The American Physical Society}