We report the observation of ${B}_{s}^{0}\mathrm{\text{\ensuremath{-}}}{\overline{B}}_{s}^{0}$ oscillations from a time-dependent measurement of the ${B}_{s}^{0}\mathrm{\text{\ensuremath{-}}}{\overline{B}}_{s}^{0}$ oscillation frequency $\ensuremath{\Delta}{m}_{s}$. Using a data sample of $1\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of $p\overline{p}$ collisions at $\sqrt{s}=1.96\text{ }\text{ }\mathrm{TeV}$ collected with the CDF II detector at the Fermilab Tevatron, we find signals of 5600 fully reconstructed hadronic ${B}_{s}$ decays, 3100 partially reconstructed hadronic ${B}_{s}$ decays, and 61 500 partially reconstructed semileptonic ${B}_{s}$ decays. We measure the probability as a function of proper decay time that the ${B}_{s}$ decays with the same, or opposite, flavor as the flavor at production, and we find a signal for ${B}_{s}^{0}\mathrm{\text{\ensuremath{-}}}{\overline{B}}_{s}^{0}$ oscillations. The probability that random fluctuations could produce a comparable signal is $8\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}8}$, which exceeds $5\ensuremath{\sigma}$ significance. We measure $\ensuremath{\Delta}{m}_{s}=17.77\ifmmode\pm\else\textpm\fi{}0.10(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.07(\mathrm{syst})\text{ }\text{ }{\mathrm{ps}}^{\ensuremath{-}1}$ and extract $|{V}_{\mathrm{td}}/{V}_{\mathrm{ts}}|=0.2060\ifmmode\pm\else\textpm\fi{}0.0007(\ensuremath{\Delta}{m}_{s}{)}_{\ensuremath{-}0.0060}^{+0.0081}(\ensuremath{\Delta}{m}_{d}+\mathrm{theor})$.