  Pseudo-Riemannian manifolds with nonzero parallel Weyl tensor which are not
locally symmetric are known as ECS manifolds. Every ECS manifold carries a
distinguished null parallel distribution $\mathcal{D}$, the rank $d \in \{ 1, 2
\}$ of which is referred to as the rank of the manifold itself. Under a natural
genericity assumption on the Weyl tensor, we fully describe the universal
coverings of compact rank-one ECS manifolds. We then show that any generic
compact rank-one ECS manifold must be translational, in the sense that the
holonomy group of the natural flat connection induced on $\mathcal{D}$ is
either trivial or isomorphic to $\mathbb{Z}_2$. We also prove that all
four-dimensional rank-one ECS manifolds are noncompact, this time without
assuming genericity, as it is always the case in dimension four.
