Proton mobility in water determines the conductive properties of water-based proton conductors. We address the problem of proton mobility in pure water using a new, simple, Newtonian molecular dynamics water model which is applicable to proton-rich environments (e.g., polymer electrolyte membranes). This model has degrees of freedom that are "inertial" and "inertialess" relative to the proton. The solvated proton is treated using a local empirical valence bond Hamiltonian, which allows for the efficient simulation of full charge, energy-conserving dynamics in single and multiple-proton systems. The solvated proton displays the Grotthus-type proton transfer mechanism, giving significantly enhanced transport in comparison with the classical diffusion of an H3O+ ion. The model yields an activation energy of 0.11 eV, in excellent agreement with experiment. The results are consistent with the observation that nonpolarizable water models, conditioned to reproduce correct values of the static dielectric constant, are predestined to give too large activation energies of proton mobility due to the overweighted spectrum of the slower nuclear modes.