Dimethyl sulfoxide (DMSO) in aqueous solution is widely used for the preservation of biological tissues under freezing conditions. DMSO and other agents are believed to act colligatively to depress the freezing point of water and, importantly, to promote the vitrification of water to prevent its crystallisation and the ensuing damage arising from the formation of intracellular ice. However, there has been no direct evidence of the precise effect of these agents on the vitrification properties of water. Here we report direct computational evidence, using molecular dynamics annealing simulations carried out within the experimentally inaccessible region in supercooled water, of a broadening of the glass transition of water, corresponding to the formation of a stronger glass in aqueous DMSO solutions. These findings provide insight at the molecular level into the mechanism of solvent cryoprotection by suggesting that the resulting thermodynamic stability of the glassy state of water reduces the probability of its nucleation and the subsequent formation of ice as the temperature is decreased.