We present an updated measurement of the ${B}_{s}^{0}$ lifetime using the semileptonic decays ${B}_{s}^{0}\ensuremath{\rightarrow}{D}_{s}^{\ensuremath{-}}{\ensuremath{\mu}}^{+}\ensuremath{\nu}X$, with ${D}_{s}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\phi}{\ensuremath{\pi}}^{\ensuremath{-}}$ and $\ensuremath{\phi}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}$ (and the charge conjugate process). This measurement uses the full Tevatron Run II sample of proton-antiproton collisions at $\sqrt{s}=1.96\text{ }\text{ }\mathrm{TeV}$, comprising an integrated luminosity of $10.4\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. We find a flavor-specific lifetime ${\ensuremath{\tau}}_{\mathrm{fs}}({B}_{s}^{0})=1.479\ifmmode\pm\else\textpm\fi{}0.010(\text{stat})\ifmmode\pm\else\textpm\fi{}0.021(\text{syst})\text{ }\text{ }\mathrm{ps}$. This technique is also used to determine the ${B}^{0}$ lifetime using the analogous ${B}^{0}\ensuremath{\rightarrow}{D}^{\ensuremath{-}}{\ensuremath{\mu}}^{+}\ensuremath{\nu}X$ decay with ${D}^{\ensuremath{-}}\ensuremath{\rightarrow}\ensuremath{\phi}{\ensuremath{\pi}}^{\ensuremath{-}}$ and $\ensuremath{\phi}\ensuremath{\rightarrow}{K}^{+}{K}^{\ensuremath{-}}$, yielding $\ensuremath{\tau}({B}^{0})=1.534\ifmmode\pm\else\textpm\fi{}0.019(\text{stat})\ifmmode\pm\else\textpm\fi{}0.021(\text{syst})\text{ }\text{ }\mathrm{ps}$. Both measurements are consistent with the current world averages, and the ${B}_{s}^{0}$ lifetime measurement is one of the most precise to date. Taking advantage of the cancellation of systematic uncertainties, we determine the lifetime ratio ${\ensuremath{\tau}}_{\mathrm{fs}}({B}_{s}^{0})/\ensuremath{\tau}({B}^{0})=0.964\ifmmode\pm\else\textpm\fi{}0.013(\text{stat})\ifmmode\pm\else\textpm\fi{}0.007(\text{syst})$.