  We compute the full holonomy group of compact Lorentzian manifolds with
parallel Weyl tensor, which are neither conformally flat nor locally symmetric,
for the case where the fundamental group is contained in a distinguished
subgroup G of the isometry group of the universal cover. To prove this, we show
that every such compact Lorentzian manifold has to be geodesically complete.
Moreover we characterize the identity component of the isometry group for this
universal cover and show that G is up to a discrete factor contained in the
latter. Concretely, we prove that the identity component of the isometry group
is isomorphic to a semidirect product of a subgroup of SO(n) with the
Heisenberg group.
