Uniformisation de l'espace des feuilles de certains feuilletages de codimension 1

This paper deals with codimension one (may be singular) foliations on compact K"alher manifolds whose conormal bundle is assumed to be pseudo-effective. Using currents with minimal singularities, we show that one can endow the space of leaves with a metric of constant non positive curvature wich may degenerate on a rigidly embedded invariant hypersurface.