The first observation of the B-s(0) -> D-s*(-/+) K-+/- decay is reported using 3.0 fb(-1) of proton-proton collision data collected by the LHCb experiment. The D-s*(-/+) mesons are reconstructed through the decay chain D-s*(-/+) -> gamma D-s(-/+) ((KK +/-)-K--/+pi(-/+)). The branching fraction relative to that for B-s(0) -> D-s*(-)pi(+) decays is measured to be B (B-s(0) -> D-sK--/+(+/-))/B(B-s(0) -> D-s(-)pi(+)) = 0.068 +/- 0.005(-0.002)(+0.003), where the first uncertainty is statistical and the second is systematic. Using a recent measurement of B(B-s(0) -> D-s*(-)pi(+)), the absolute branching fraction of B-s(0) -> DsK--/+(+/-) is measured as B(B-s(0) -> DK--/+(+/-)) = (16.3 +/- 1.2(stat)(-0.5)(+0.7)(syst) +/- 4.8(norm)) x 10(-5), where the third uncertainty is due to the uncertainty on the branching fraction of the normalisation channel.