  In this paper we provide a positive answer to a conjecture due to A. J. Di
Scala, A. Loi, H. Hishi (see [3, Conjecture 1]) claiming that a
simply-connected homogeneous K\"ahler manifold M endowed with an integral
K\"ahler form $\mu\omega$, admits a holomorphic isometric immersion in the
complex projective space, for a suitable $\mu>0$. This result has two
corollaries which extend to homogeneous K\"ahler manifolds the results obtained
by the authors in [8] and in [12] for homogeneous bounded domains.
