Using data from the Fermilab photoproduction experiment E691, we have measured the branching ratio for the decay ${D}^{0}\ensuremath{\rightarrow}{\overline{K}}^{0}{\ensuremath{\pi}}^{0}$ to be (5.0 \ifmmode\pm\else\textpm\fi{} 0.8 \ifmmode\pm\else\textpm\fi{} 0.9)%. We see no evidence for the decay ${D}^{0}\ensuremath{\rightarrow}{\overline{K}}^{0}{K}^{0}$ and set a 90% confidence level upper limit of 0.12%. The large ${\overline{K}}^{0}{\ensuremath{\pi}}^{0}$ branching ratio (relative to ${K}^{\ensuremath{-}}{\ensuremath{\pi}}^{+}$) indicates either a lack of color suppression or significant elastic final-state interactions. The upper limit on the ${\overline{K}}^{0}{K}^{0}$ mode indicates that final-state interactions mix final states such as ${K}^{+}{K}^{\ensuremath{-}}$ or ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ into the ${\overline{K}}^{0}{K}^{0}$ less than 27% of the time at the 90% confidence level.