We study water and dimethyl-sulfoxide (DMSO) dynamics in the eutectic DMSO/water mixture, 2H(2)O:1DMSO, by incoherent quasielastic and inelastic neutron scattering. A temperature range from room down to -27 degreesC is investigated. Both water and DMSO translational dynamics are significantly slowed in the mixture compared to pure liquids. They exhibit different diffusion dynamics, pointing to the absence of stable hydrogen bonded complexes. Further, the presence of a solute suppresses the dynamic anomalies of water: self-diffusion coefficients for water in the mixture, as well as residence times for jump diffusion reveal an Arrhenius temperature dependence, in contrast to the strong non-Arrhenius behavior in pure water. The density of vibrational states shows a shift of the characteristic librational peak in pure water towards higher energies in the mixture, reflecting hindrance of large amplitude proton motions. These results are compatible with computer simulations which predict longer hydrogen-bond lifetimes in the mixture, compared to pure water. However, rotational relaxation times for water protons in the mixture are estimated to be similar to1 ps, as for pure water. This experimental finding suggests that unlike in pure, low temperature water, proton librations should no longer be the main mechanism responsible for hydrogen-bond breaking in the mixture. Thus, the correspondence between the proton hindered rotational relaxation time and the "hydrogen-bond lifetime" should not apply in DMSO-water mixtures.