The water wave problem models the free-surface evolution of an ideal fluid under the influence of gravity and surface tension. The governing equations are a central model in the study of open ocean wave propagation, but they possess a surprisingly difficult and subtle well-posedness theory. In this paper we establish the existence and uniqueness of solutions to the water wave equations augmented with physically inspired viscosity suggested in the recent work of Dias, Dyachenko, and Zakharov (Phys. Lett. A, 372, 2008). As we show, this viscosity (which can be arbitrarily weak) not only delivers an enormously simplified well-posedness theory for the governing equations, but also justifies a greatly stabilized numerical scheme for use in studying solutions of the water wave problem.