  In this paper, we build a Gibbs measure for the cubic defocusing
Schr\"odinger equation on the real line with a decreasing interaction
potential, in the sense that the non linearity $|u|^2u$ is multiplied by a
function $\chi$ which we assume integrable and smooth enough. We prove that
this equation is globally well-posed in the support of this measure and that
the measure is invariant under the flow of the equation. What is more, the
support of the measure (the set of initial data) is disjoint from $L^2$.
