Consider two stochastically independent, stationary Gaussian lattice processes with zero means, {X(u), u (Z2} and {Y(u), u (Z2}. An asymptotic expression for the variance of the sample correlation between {X(u)} and {Y(u)} over a finite square is derived. This expression also holds for a wide class of domains in Z2. As an illustration, the asymptotic variance of the correlation between two first-order autonormal schemes is evaluated.