We present measurements of the ${\ensuremath{\Lambda}}_{b}^{0}$ lifetime in the exclusive decay channel ${\ensuremath{\Lambda}}_{b}^{0}\ensuremath{\rightarrow}J/\ensuremath{\psi}{\ensuremath{\Lambda}}^{0}$, with $J/\ensuremath{\psi}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$ and ${\ensuremath{\Lambda}}^{0}\ensuremath{\rightarrow}p{\ensuremath{\pi}}^{\ensuremath{-}}$, the ${B}^{0}$ lifetime in the decay ${B}^{0}\ensuremath{\rightarrow}J/\ensuremath{\psi}{K}_{S}^{0}$ with $J/\ensuremath{\psi}\ensuremath{\rightarrow}{\ensuremath{\mu}}^{+}{\ensuremath{\mu}}^{\ensuremath{-}}$ and ${K}_{S}^{0}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$, and the ratio of these lifetimes. The analysis is based on approximately $250\text{ }\text{ }{\mathrm{pb}}^{\ensuremath{-}1}$ of data recorded with the D0 detector in $p\overline{p}$ collisions at $\sqrt{s}=1.96\text{ }\text{ }\mathrm{TeV}$. The ${\ensuremath{\Lambda}}_{b}^{0}$ lifetime is determined to be $\ensuremath{\tau}({\ensuremath{\Lambda}}_{b}^{0})={1.22}_{\ensuremath{-}0.18}^{+0.22}(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.04(\mathrm{syst})\text{ }\text{ }\mathrm{ps}$, the ${B}^{0}$ lifetime $\ensuremath{\tau}({B}^{0})={1.40}_{\ensuremath{-}0.10}^{+0.11}(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.03(\mathrm{syst})\text{ }\text{ }\mathrm{ps}$, and the ratio $\ensuremath{\tau}({\ensuremath{\Lambda}}_{b}^{0})/\ensuremath{\tau}({B}^{0})={0.87}_{\ensuremath{-}0.14}^{+0.17}(\mathrm{stat})\ifmmode\pm\else\textpm\fi{}0.03(\mathrm{syst})$. In contrast with previous measurements using semileptonic decays, this is the first determination of the ${\ensuremath{\Lambda}}_{b}^{0}$ lifetime based on a fully reconstructed decay channel.