  In an earlier paper the authors provided general conditions on a real
codimension 2 submanifold $S\subset C^{n}$, $n\ge 3$, such that there exists a
possibly singular Levi-flat hypersurface $M$ bounded by $S$.
  In this paper we consider the case when $S$ is a graph of a smooth function
over the boundary of a bounded strongly convex domain $\Omega\subset
C^{n-1}\times R$ and show that in this case $M$ is necessarily a graph of a
smooth function over $\Omega$. In particular, $M$ is non-singular.
