We exploit the potential of a combined dielectric spectroscopy (DS) and deuteron nuclear magnetic resonance (H-2 NMR) approach to investigate the molecular dynamics in a supercooled 2:1 molar mixture of deuterated water (D2O) and dimethyl sulfoxide (DMSO). While DS probes the rotational motion of both components, application of H-2 NMR allows us to single out the dynamical behavior of the water molecules. Combining the results of both methods, we can follow the slowdown of the alpha-process of the mixture over more than 10 orders of magnitude in time, revealing that the Vogel-Fulcher-Tamrnann (VFT) equation describes well its temperature dependence down to the glass transition temperature, T-g = 146 K. While the H-2 NMR data do not provide evidence for a secondary relaxation process in the weakly supercooled regime, they indicate that, in the deeply supercooled regime, T-g <= T <= 160 K, the water molecules do show a secondary dynamical process, which is faster and exhibits a weaker temperature dependence than the a-process of the mixture. Consistently, the shape of the dielectric spectra changes in this temperature range. H-2 NMR rotational correlation functions reveal that this faster secondary water process destroys essentially all orientational correlation. In addition, these data show that the water reorientation process is characterized by a mean elementary jump angle smaller than 13 degrees. Possible origins of the faster secondary water process in the deeply supercooled mixture are discussed.