We find, by means of a deep inelastic neutron scattering experiment, a significant excess of proton mean kinetic energy E_(k) in supercooled water, compared with that measured in stable liquid and solid phases. The measured values of E_(k) at moderate degrees of supercooling do not fit the predicted linear increase with temperature observed for the water stable phases. This anomalous behavior is confirmed by the shape of the measured momentum distribution, thus supporting a likely occurrence of ground-state quantum delocalization of a proton between the O atoms of two neighboring molecules. These results strongly suggest a transition from a single-well to a double-well potential felt by the delocalized proton, with a reduced first neighbor O-O distance, in the supercooled state, as compared to ambient condition.