Within the classical field model, we find that the phase of a uniform Bose-Einstein condensate at nonzero temperature undergoes a true diffusive motion in the microcanonical ensemble, the variance of the condensate phase change between time zero and time
t growing linearly in
t. The phase diffusion coefficient obeys a simple scaling law in the double thermodynamic and Bogoliubov limit. We construct an approximate calculation of the diffusion coefficient, in fair agreement with the numerical results over the considered temperature range, and we extend this approximate calculation to the quantum field.