The semileptonic and nonleptonic decays of the double-beauty axial-vector tetraquark T-bb;(u) over bar(d) over bar(-) to a state (T) over tilde (0)(bc;(u) over bar(d) over bar) (hereafter T-bb(-) and (T) over tilde (0)(bc), respectively) are investigated in the context of the QCD sum rule method. The final-state tetraquark (T) over tilde (0)(bc) is treated as an axial-vector particle built of a heavy axial-vector diquark b(T)gamma C-mu(c) and light scalar antidiquark (u) over barC(gamma 5)(d) over bar (T). Its spectroscopic parameters are calculated using the two-points sum rules by taking into account contributions of quark, gluon and mixed condensates up to dimension 10. We study the dominant semileptonic T-bb(-) -> (T) over tilde (0)(bc)l (nu) over bar (l) and nonleptonic decays T-bb(-) -> (T) over tilde M-0(bc), where M is one of the pseudoscalar mesons pi(-), K-, D- and Ds(-). The partial widths of these processes are computed in terms of weak form factors G(i)(q(2)), i = 1, 2, 3, 4, extracted by employing the QCD three-point sum rule approach. Predictions obtained for partial widths of considered decays are used to improve accuracy of theoretical predictions for full width and lifetime of the tetraquark T-bb(-), which are important for experimental exploration of this exotic meson.