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The first observation of the suppressed semileptonic $B_s^0 \toK^-\mu^+\nu_\mu$ decay is reported. Using a data sample recorded in {\it pp}collisions in 2012 with the LHCb detector, corresponding to an integratedluminosity of 2 $\mathrm{fb}^{-1}$, the branching fraction\mbox{$\mathcal{B}(B_s^0 \to K^-\mu^+\nu_\mu)$} is measured to be$(1.06\pm0.05~(\mathrm{stat})\pm0.08~(\mathrm{syst}))\times 10^{-4}$, where thefirst uncertainty is statistical and the second one represents the combinedsystematic uncertainties. The decay $B_s^0 \to D_s^-\mu^+\nu_\mu$, where$D_s^-$ is reconstructed in the final state $K^+K^-\pi^-$, is used as anormalization channel to minimize the experimental systematic uncertainty.Theoretical calculations on the form factors of the $B_s^0 \to K^-$ and $B_s^0\to D_s^-$ transitions are employed to determine the ratio of the CKM matrixelements ${|V_{ub}|}/{|V_{cb}|}$ at low and high $B_s^0 \to K^-$ momentumtransfer.

First observation of the decay $B_s^0 \to K^-\mu^+\nu_\mu$ and a measurement of $|V_{ub}|/|V_{cb}|$