Given G = (V,E) an undirected graph and two specified nonadjacent nodes a and b of V, a cut separator is a subset F =delta (C) E such that a,b are in V / C and a and b belong to different connected components of the graph induced by V / C. Given a non-negative cost vector, the cut separator problem is to find a cut separator of minimum cost. This new problem can be seen as a generalization of the vertex separator problem. In this article, we give a polynomial time algorithm for this problem. We also present six equivalent linear programming formulations, and we show their tightness. Using these results we obtain an explicit short polyhedral description of the dominant of the cut separator polytope.