From an analysis of the decay ${B}_{s}^{0}\ensuremath{\rightarrow}J/\ensuremath{\psi}\ensuremath{\phi}$, we obtain the width difference between the light and heavy mass eigenstates $\ensuremath{\Delta}\ensuremath{\Gamma}\ensuremath{\equiv}({\ensuremath{\Gamma}}_{L}\ensuremath{-}{\ensuremath{\Gamma}}_{H})=0.17\ifmmode\pm\else\textpm\fi{}0.09(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.02(\mathrm{syst})\text{ }\text{ }{\mathrm{ps}}^{\ensuremath{-}1}$ and the $CP$-violating phase ${\ensuremath{\phi}}_{s}=\ensuremath{-}0.79\ifmmode\pm\else\textpm\fi{}0.56(\mathrm{stat}{)}_{\ensuremath{-}0.01}^{+0.14}(\mathrm{syst})$. Under the hypothesis of no $CP$ violation (${\ensuremath{\phi}}_{s}\ensuremath{\equiv}0$), we obtain $1/\overline{\ensuremath{\Gamma}}=\phantom{\rule{0ex}{0ex}}\overline{\ensuremath{\tau}}({B}_{s}^{0})=1.52\ifmmode\pm\else\textpm\fi{}0.08(\mathrm{stat}{)}_{\ensuremath{-}0.03}^{+0.01}(\mathrm{syst})\text{ }\text{ }\mathrm{ps}$ and $\ensuremath{\Delta}\ensuremath{\Gamma}={0.12}_{\ensuremath{-}0.10}^{+0.08}(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.02(\mathrm{syst})\text{ }\text{ }{\mathrm{ps}}^{\ensuremath{-}1}$. The data sample corresponds to an integrated luminosity of about $1.1\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ accumulated with the D0 detector at the Fermilab Tevatron collider. This is the first direct measurement of the $CP$-violating mixing phase in the ${B}_{s}^{0}$ system.