The first observation of the ${B}_{s}^{0}\ensuremath{\rightarrow}{\overline{D}}^{0}{K}_{S}^{0}$ decay mode and evidence for the ${B}_{s}^{0}\ensuremath{\rightarrow}{\overline{D}}^{*0}{K}_{S}^{0}$ decay mode are reported. The data sample corresponds to an integrated luminosity of $3.0\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ collected in $pp$ collisions by LHCb at center-of-mass energies of 7 and 8 TeV. The branching fractions are measured to be $\mathcal{B}({B}_{s}^{0}\ensuremath{\rightarrow}{\overline{D}}^{0}{\overline{K}}^{0})=[4.3\ifmmode\pm\else\textpm\fi{}0.5(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.3(\mathrm{syst})\ifmmode\pm\else\textpm\fi{}0.3(\mathrm{frag})\ifmmode\pm\else\textpm\fi{}0.6(\mathrm{norm})]\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4},\phantom{\rule{0ex}{0ex}}\mathcal{B}({B}_{s}^{0}\ensuremath{\rightarrow}{\overline{D}}^{*0}{\overline{K}}^{0})=[2.8\ifmmode\pm\else\textpm\fi{}1.0(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.3(\mathrm{syst})\ifmmode\pm\else\textpm\fi{}0.2(\mathrm{frag})\ifmmode\pm\else\textpm\fi{}0.4(\mathrm{norm})]\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4},$where the uncertainties are due to contributions coming from statistical precision, systematic effects, and the precision of two external inputs, the ratio ${f}_{s}/{f}_{d}$ and the branching fraction of ${B}^{0}\ensuremath{\rightarrow}{\overline{D}}^{0}{K}_{S}^{0}$, which is used as a calibration channel.