  Pseudo-Riemannian manifolds with parallel Weyl tensor that are not
conformally flat or locally symmetric, also known as ECS manifolds, have a
natural local invariant, the rank, which equals 1 or 2, and is the dimension of
a certain distinguished null parallel distribution $\,\mathcal{D}$. All known
examples of compact ECS manifolds are of rank one and have dimensions greater
than 4. We prove that a compact rank-one ECS manifold, if not locally
homogeneous, replaced when necessary by a two-fold isometric covering, must be
a bundle over the circle with leaves of $\,\mathcal{D}^\perp$ serving as the
fibres. The same conclusion holds in the locally-homogeneous case if one
assumes that $\,\mathcal{D}^\perp$ has at least one compact leaf. We also show
that in the pseudo-Riemannian universal covering space of any compact rank-one
ECS manifold the leaves of $\,\mathcal{D}^\perp$ are the factor manifolds of a
global product decomposition.
