The first observation of the ${B}_{s}^{0}\ensuremath{\rightarrow}{\ensuremath{\eta}}^{\ensuremath{'}}{\ensuremath{\eta}}^{\ensuremath{'}}$ decay is reported. The study is based on a sample of proton-proton collisions corresponding to $3.0\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of integrated luminosity collected with the LHCb detector. The significance of the signal is 6.4 standard deviations. The branching fraction is measured to be $[3.31\ifmmode\pm\else\textpm\fi{}0.64(\text{stat})\ifmmode\pm\else\textpm\fi{}0.28(\text{syst})\ifmmode\pm\else\textpm\fi{}0.12(\text{norm})]\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}5}$, where the third uncertainty comes from the ${B}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{\ensuremath{\eta}}^{\ensuremath{'}}{K}^{\ifmmode\pm\else\textpm\fi{}}$ branching fraction that is used as a normalization. In addition, the charge asymmetries of ${B}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{\ensuremath{\eta}}^{\ensuremath{'}}{K}^{\ifmmode\pm\else\textpm\fi{}}$ and ${B}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}\ensuremath{\phi}{K}^{\ifmmode\pm\else\textpm\fi{}}$, which are control channels, are measured to be $(\ensuremath{-}0.2\ifmmode\pm\else\textpm\fi{}1.3)%$ and $(+1.7\ifmmode\pm\else\textpm\fi{}1.3)%$, respectively. All results are consistent with theoretical expectations.