  An n-dimensional strictly pseudoconvex Hartogs domain D_F can be equipped
with a natural Kaehler metric g_F. In this paper we prove that if m_0g_F is
balanced for a given positive integer m_0 then m_0>n and (D_F, g_F) is
holomorphically isometric to an open subset of the n-dimensional complex
hyperbolic space.
