  Let $\cal{F}$ be a regular codimension 1 holomorphic foliation on a compact
K\" ahler manifold. One assumes in addition that $\cal{F}$ possesses a
transverse invariant positive current. The aim of this paper is to establish
the following alternative: - There exists an invariant hypersurface. - The
foliation admits a transverse invariant hermitian metric with constant
curvature
