  We prove that a K\"ahler group which is cubulable, i.e. which acts properly
discontinuously and cocompactly on a CAT(0) cubical complex, has a finite index
subgroup isomorphic to a direct product of surface groups, possibly with a free
Abelian factor. Similarly, we prove that a closed aspherical K\"ahler manifold
with a cubulable fundamental group has a finite cover which is biholomorphic to
a topologically trivial principal torus bundle over a product of Riemann
surfaces. Along the way, we prove a factorization result for essential actions
of K\"ahler groups on irreducible, locally finite CAT(0) cubical complexes,
under the assumption that there is no fixed point in the visual boundary.
