  Let $\alpha$ be a polygonal Jordan curve in $\bfR^3$. We show that if
$\alpha$ satisfies certain conditions, then the least-area Douglas-Rad\'{o}
disk in $\bfR^3$ with boundary $\alpha$ is unique and is a smooth graph. As our
conditions on $\alpha$ are not included amongst previously known conditions for
embeddedness, we are enlarging the set of Jordan curves in $\bfR^3$ which are
known to be spanned by an embedded least-area disk. As an application, we
consider the conjugate surface construction method for minimal surfaces. With
our result we can apply this method to a wider range of complete catenoid-ended
minimal surfaces in $\bfR^3$.
