The lifetime of the $B_s$ meson is measured using the semileptonic decay $B_s \to D_s^- \ell^+ \nu X$. The data sample consists of 19.3 pb$^{-1}$ of $p \bar{p}$ collisions at $\sqrt{s} = 1.8$ TeV collected by the CDF detector at the Fermilab Tevatron collider during 1992-1993. There are $76 \pm 8$ $\ell^{+} D_s^{-}$ signal events where the $D_s$ is identified via the decay $D_s^{-} \to \phi \pi^{-}$,~$\phi \to K^+ K^-$. Using these events, the $B_s$ meson lifetime is determined to be $\tau_{s} = 1.42^{+ 0.27}_{- 0.23} {\rm (stat)} \pm 0.11 {\rm (syst)}$ ps. A measurement of the $B_s$ lifetime in a low statistics sample of exclusive $B_s \to J/\psi \phi$ decays is also presented in this paper.