First measurements of the self-dynamics of liquid water in the GPa range are reported. The GPa range has here become accessible through a new setup for the Paris-Edinburgh press specially conceived for quasielastic neutron scattering studies. A direct measurement of both the translational and rotational diffusion coefficients of water along the 400 K isotherm up to 3 GPa, corresponding to the melting point of ice VII, is provided and compared with molecular dynamics simulations. The translational diffusion is observed to strongly decrease with pressure, though its variation slows down for pressures higher than 1 GPa and decouples from that of the shear viscosity. The rotational diffusion turns out to be insensitive to pressure. Through comparison with structural data and molecular dynamics simulations, we show that this is a consequence of the rigidity of the first neighbors shell and of the invariance of the number of hydrogen bonds of a water molecule under high pressure. These results show the inadequacy of the Stokes-EinsteinDebye equations to predict the self-diffusive behavior of water at high temperature and high pressure, and challenge the usual description of hot dense water behaving as a simple liquid. DOI: 10.1103/PhysRevLett.111.185901 PACS numbers: 66.10.CÀ, 45.20.dc, 62.50.Àp, 78.70.Nx Water is, by far, the most well-known substance in nature, as essentially most of its properties have been measured with high accuracy [1], at least at atmospheric or moderate pressure. Water under high pressure (HP) has also been the subject of several investigations [2-13], as it affects fields ranging from condensed matter physics to planetary science. However, while structural [2] and vibrational properties [3] of water have been explored up to several GPa, the knowledge of water self-dynamics under high pressure lags far behind [14][15][16][17]. Information on water diffusion under HP conditions is of paramount importance for several issues in applied and fundamental science. As examples, the diffusion of water at pressures of few GPa, typical of the transition zone of the Earth's mantle, has a strong incidence on the processes governing intermediate-depth seismicity. New phases of water characterized by peculiar diffusive behavior, as free rotation of water molecules [12] (plastic phases), or proton free diffusion [13] (superionic phase) are predicted by computer simulations [2-4] at pressures of several GPa. Their characterization is essential in order to develop reliable models of planetary interiors.We developed a new setup for the Paris-Edinburgh (PE) press [18,19] adapted to quasielastic incoherent neutron scattering (QENS) measurements [20], giving access to hydrogen diffusion and water molecular reorientation in the GPa range (up to 5 GPa and 550 K). Incoherent neutron scattering is a unique probe of individual motions of atoms and, in particular, of hydrogen, due to its huge incoherent cross section, typically 2 orders of magnitude larger than in other elements. Applied to water, QENS measures the t...