  Let $M$ be a simply connected pseudo-Riemannian homogeneous space of finite
volume with isometry group $G$. We show that $M$ is compact and that the
solvable radical of $G$ is abelian and the Levi factor is a compact semisimple
Lie group acting transitively on $M$. For metric index less than three, we find
that the isometry group of $M$ is compact itself. Examples demonstrate that $G$
is not necessarily compact for higher indices. To prepare these results, we
study Lie algebras with abelian solvable radical and a nil-invariant symmetric
bilinear form. For these, we derive an orthogonal decomposition into three
distinct types of metric Lie algebras.
