Abstract: This paper reports the observation of the decays B-(s)(0) -> D-s1(2536)K--/+(+/-) using proton-proton collision data collected by the LHCb experiment, corresponding to an integrated luminosity of 9 fb(-1). The branching fractions of these decays are measured relative to the normalisation channel B-0 -> (D) over bar (K+K-)-K-0. The D-s1(2536)(-) meson is reconstructed in the (D) over bar*(2007)K-0(-) decay channel and the products of branching fractions are measured to be B(B-s(0) -> D-s1(2536)K--/+(+/-)) x B(D-s1(2536)(-) -> D ($) over bar*(2007)K-0(-)) = (2.49 +/- 0.11 +/- 0.12 +/- 0.25 +/- 0.06) x 10(-5), B(B-0 -> D-s1(2536)K--/+(+/-) ) x B(D-s1(2536)(-) -> (D) over bar*(2007)K-0(-)) = (0.510 +/- 0.021 +/- 0.036 +/- 0.050) x 10(-5). The first uncertainty is statistical, the second systematic, and the third arises from the uncertainty of the branching fraction of the B-0 -> (D) over bar (K+K-)-K-0 normalisation channel. The last uncertainty in the B-s(0) result is due to the limited knowledge of the fragmentation fraction ratio, f(s)/f(d). The significance for the B-s(0) and B-0 signals is larger than 10 sigma. The ratio of the helicity amplitudes which governs the angular distribution of the D-s1(2536)(-) -> (D) over bar*(2007)K-0(-) decay is determined from the data. The ratio of the S- and D-wave amplitudes is found to be 1.11 +/- 0.15 +/- 0.06 and the phase difference between them 0.70 +/- 0.09 +/- 0.04 rad, where the first uncertainty is statistical and the second systematic.