The mass and coupling of the axial-vector tetraquark T-bc; (u) over bar(d) over bar(0) (in a short form T-bc(0)) are calculated by means of the QCD two-point sum rule method. In computations we take into account contributions arising from various quark, gluon and mixed vacuum condensates up to dimension 10. The central value of the mass m = (7105 +/- 155) MeV lies below the thresholds for the strong and electromagnetic decays of the T-bc(0) state, and hence it transforms to conventional mesons only through the weak decays. In the case of m = 7260 MeV the tetraquark T-bc(0) becomes the strong- and electromagnetic-interaction unstable particle. In the first case, we find the full width and mean lifetime of T-bc(0) using its semileptonic decays T-bc(0) -> T(cc; (u) over bar(d) over bar)(+)l (nu) over bar (l) (l = e, mu, tau), where the final-state tetraquark is a scalar state. We compute also partial widths of cc;rid the nonleptonic weak decays T-bc(0) -> T-cc;(u) over bar(d) over bar(+) pi(-) (K-, D-, D-s(-)), and take into account their effects on the full width of T-bc(0). In the context of the second scenario we calculate partial widths of S-wave strong decays T-bc(0) -> B*D--(+) and T-bc(0) -> (B) over bar*(0) D-0, and using these channels evaluate the full width of T-bc(0). Predictions for Gamma(full) = (3.98 +/- 0.51) x 10(-10) MeV and mean lifetime tau = 1.65(-0.18)(+0.25) ps of T-bc(0) obtained in the context of the first option, as well as the full width Gamma(full) = (63.5 +/- 8.9) MeV extracted in the second scenario may be useful for experimental and theoretical exploration of double-heavy exotic mesons. (C) 2019 The Author(s). Published by Elsevier B.V.