  We determine the local structure of all pseudo-Riemannian manifolds $(M,g)$
in dimensions $n\ge4$ whose Weyl conformal tensor $W$ is parallel and has rank
1 when treated as an operator acting on exterior 2-forms at each point. If one
fixes three discrete parameters: the dimension $n\ge4$, the metric signature
$--...++$, and a sign factor $\epsilon=\pm1$ accounting for semidefiniteness of
$W$, then the local-isometry types of our metrics $g$ correspond bijectively to
equivalence classes of surfaces $\varSigma$ with equiaffine projectively flat
torsionfree connections; the latter equivalence relation is provided by
unimodular affine local diffeomorphisms. The surface $\varSigma$ arises,
locally, as the leaf space of a codimension-two parallel distribution on $M$,
naturally associated with $g$. We exhibit examples in which the leaves of the
distribution form a fibration with the total space $M$ and base $\varSigma$,
for a closed surface $\varSigma$ of any prescribed diffeomorphic type.
  Our result also completes a local classification of pseudo-Riemannian metrics
with parallel Weyl tensor that are neither conformally flat nor locally
symmetric: for those among such metrics which are not Ricci-recurrent, rank $W$
= 1, and so they belong to the class mentioned above; on the other hand, the
Ricci-recurrent ones have already been classified by the second author.
