The observation of the decay $\varXi^{-}_{b}\to J/\psi\varLambda K^{-}$ is reported, using a data sample corresponding to an integrated luminosity of $3~\mathrm{fb}^{-1}$, collected by the LHCb detector in $pp$ collisions at centre-of-mass energies of $7$ and $8~\mathrm{TeV}$. The production rate of $\varXi_{b}^{-}$ baryons detected in the decay $\varXi_{b}^{-}\to J/\psi\varLambda K^{-}$ is measured relative to that of $\varLambda_{b}^{0}$ baryons using the decay $\varLambda_{b}^{0}\to J/\psi \varLambda$. Integrated over the $b$-baryon transverse momentum $p_{\rm T}<25~\mathrm{GeV/}c $ and rapidity $2.0<y<4.5$, the measured ratio is \begin{equation*} \frac{f_{\varXi_{b}^{-}}}{f_{\varLambda_{b}^{0}}}\frac{\mathcal{B}(\varXi_{b}^{-}\to J/\psi\varLambda K^{-})}{\mathcal{B}(\varLambda_{b}^{0}\to J/\psi \varLambda)}=(4.19\pm 0.29~(\mathrm{stat})\pm0.15~(\mathrm{syst}))\times 10^{-2}, \end{equation*}where $f_{\varXi_{b}^{-}}$ and $f_{\varLambda_{b}^{0}}$ are the fragmentation fractions of $b\to\varXi_{b}^{-}$ and $b\to\varLambda_{b}^{0}$ transitions, and $\mathcal{B}$ represents the branching fraction of the corresponding $b$-baryon decay. The mass difference between $\varXi_{b}^{-}$ and $\varLambda_{b}^{0}$ baryons is measured to be \begin{equation*} M(\varXi_{b}^{-})-M(\varLambda_{b}^{0})=177.08\pm0.47~(\mathrm{stat})\pm0.16~(\mathrm{syst} )~\mathrm{MeV/}c^{2}. \end{equation*}