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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 ourconditions on $\alpha$ are not included amongst previously known conditions forembeddedness, we are enlarging the set of Jordan curves in $\bfR^3$ which areknown to be spanned by an embedded least-area disk. As an application, weconsider the conjugate surface construction method for minimal surfaces. Withour result we can apply this method to a wider range of complete catenoid-endedminimal surfaces in $\bfR^3$.

On embeddedness of area-minimizing disks, and an application to constructing complete minimal surfaces