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The ground states of the one-dimensional Falicov-Kimball model are investigated in the small-coupling limit, using nearly degenerate perturbation theory. For rational electron and ion densities, respectively, equal to p/q, pi/q, with p relatively prime to q and pi/q close enough to 1/2, we find that in the ground state the ion configuration has a period q. The situation is analogous to the Peierls instability, where the usual arguments predict a period-q state that produces a gap at the Fermi level and is insulating. However for pi/q far enough from 1/2, this phase becomes unstable against phase separation. The ground state is a mixture of a period-q ionic configuration and an empty (or full) configuration, where both configurations have the same electron density to leading order. Combining these results with those previously obtained for strong coupling, it follows that a phase transition occurs in the ground state, as a function of the coupling, for ion densities far enough from 1/2.