  We solve the Levi-flat Plateau problem in the following case. Let $M \subset
{\mathbb C}^{n+1}$, $n \geq 2$, be a connected compact real-analytic
codimension-two submanifold with only nondegenerate CR singularities. Suppose
$M$ is a diffeomorphic image via a real-analytic CR map of a real-analytic
hypersurface in ${\mathbb C}^n \times {\mathbb R}$ with only nondegenerate CR
singularities. Then there exists a unique compact real-analytic Levi-flat
hypersurface, nonsingular except possibly for self-intersections, with boundary
$M$. We also study boundary regularity of CR automorphisms of domains in
${\mathbb C}^n \times {\mathbb R}$.
