  The spectroscopic parameters and decay channels of the axial-vector
tetraquark $T_{bb;\overline{u}\overline{s}}^{-}$ (in what follows, $T_{b:
\overline{s}}^{\mathrm{AV}}$) are explored using the quantum chromodynamics
(QCD) sum rule method. The mass and coupling of this state are calculated using
two-point sum rules by taking into account various vacuum condensates, up to 10
dimensions. Our prediction for the mass of this state $m=(10215\pm 250)~
\mathrm{MeV}$ confirms that it is stable with respect to strong and
electromagnetic decays and can dissociate to conventional mesons only via weak
transformations. We investigate the dominant semileptonic $T_{b:\overline{s}
}^{\mathrm{AV}} \to \mathcal{Z}_{b:\overline{s}}^{0}l\overline{\nu}_l$ and
nonleptonic $T_{b:\overline{s}}^{\mathrm{AV}} \to \mathcal{Z}_{b:\overline{s}
}^{0}M$ decays of $T_{b:\overline{s}}^{\mathrm{AV}}$. In these processes, $
\mathcal{Z}_{b:\overline{s}}^{0}$ is a scalar tetraquark $[bc][\overline{u}
\overline{s}]$ built of a color-triplet diquark and an antidiquark, whereas $M$
is one of the vector mesons $\rho ^{-}$, $K^{\ast}(892)$, $D^{\ast
}(2010)^{-}$, and $D_{s}^{\ast -}$. To calculate the partial widths of these
decays, we use the QCD three-point sum rule approach and evaluate the weak
transition form factors $G_{i}$ $(i=0,1,2,3)$, which govern these processes.
The full width $\Gamma _{\mathrm{full}} =(12.9\pm 2.1)\times
10^{-8}~\mathrm{MeV}$ and the mean lifetime $
\tau=5.1_{-0.71}^{+0.99}~\mathrm{fs}$ of the tetraquark $T_{b:\overline{s}}^{
\mathrm{AV}}$ are computed using the aforementioned weak decays. The obtained
information about the parameters of $T_{b:\overline{s}}^{\mathrm{AV}}$ and $
\mathcal{Z}_{b:\overline{s}}^{0}$ is useful for experimental investigations of
these double-heavy exotic mesons.
