Consider a lattice gas evolving according to the conservative Kawasaki dynamics at inverse temperature β on a two dimensional torus ΛL = {0, . . . , L - 1}2. We prove the tunneling behavior of the process among the states of minimal energy. More precisely, assume that there are n2 particles, n < L/2, and that the initial state is the configuration in which all sites of the square {0, . . . , n - 1}2 are occupied. We show that in the time scale e2β the process evolves as a Markov process on ΛL which jumps from any site x to any other site y ≠ x at a strictly positive rate which can be expressed in terms of the hitting probabilities of simple Markovian dynamics.