Supersymmetry breaking with vanishing cosmological constant in string theory

We show that for the effective four-dimensional N=1 supergravity governing string theory around a large class of vacua the scalar potential is the sum of non-negative terms even in the presence of supersymmetry breaking. It is therefore naturally minimized with vanishing cosmological constant. Our result relies only on generic features of these vacua that have been proven for string tree level and at all orders in σ-model perturbation theory, but appear to apply more generally, for example, holding exactly in the σ-model couplings for (0,2) ZN orbifold models of rank 16.