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We consider the Kawasaki dynamics at inverse temperature $\beta$ for theIsing lattice gas on a two-dimensional square of length $2L+1$ with periodicboundary conditions. We assume that initially the particles form a square oflength $n$, which may increase, as well as $L$, with $\beta$. We show that in aproper time scale the particles form almost always a square and that the centerof mass of the square evolves as a Brownian motion when the temperaturevanishes.

Zero-temperature limit of the Kawasaki dynamics for the Ising lattice gas in a large two-dimensional torus