Abstract: The first observation of the suppressed semileptonic B-s(0) -> K-mu(+)nu(mu) decay is reported. Using a data sample recorded in pp collisions in 2012 with the LHCb detector, corresponding to an integrated luminosity of 2 fb(-1), the branching fraction B(B-s(0) -> K-mu(+)nu(mu)) is measured to be [1.06 +/- 0.05(stat) +/- 0.08(syst)] x 10(-4), where the first uncertainty is statistical and the second one represents the combined systematic uncertainties. The decay B-s(0) -> D-s(-)mu(+)nu(mu), where D-s(-) is reconstructed in the final state K+K-pi(-), is used as a normalization channel to minimize the experimental systematic uncertainty. Theoretical calculations on the form factors of the B-s(0) -> K- and B-s(0) -> D-s(-) transitions are employed to determine the ratio of the Cabibbo-Kobayashi-Maskawa matrix elements vertical bar V-ub vertical bar/vertical bar V-cb vertical bar at low and high B-s(0) -> K- momentum transfer.