We consider a new cellular automata rule for a synchronous random walk on a two-dimensional square lattice, subject to an exclusion principle. It is found that the macroscopic behavior of our model obeys the telegraphists's equation, with an adjustable diffusion constant. By construction, the dynamics of our
model is exactly described by a linear discrete Boltzmann equation which is solved analytically for some boundary conditions. Consequently, the connection between the microscopic and the macroscopic descriptions is obtained exactly and the continuous limit studied rigorously. The typical system size for which a true diffusive behavior is observed may be deduced as a function of the parameters entering into the rule.