  The mass and coupling of the axial-vector tetraquark $T_{bc;\bar{u}\bar{d}
}^{0}$ (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 $m=(7105 \pm 155)~\mathrm{ MeV}$ lies below
the thresholds for the strong and electromagnetic decays of $T_{bc}^{0}$ state,
and hence it transforms to conventional mesons only through the weak decays. In
the case of $m=7260~\mathrm{MeV}$ 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 dominant
semileptonic decays $T_{bc}^{0}\to T_{cc;\bar{ u}\bar{d}}^{+}l\overline{\nu
}_{l}$ ($l=e,\ \mu, \tau$), where the final-state tetraquark is a scalar state.
We compute also partial widths of the nonleptonic weak decays $T_{bc}^{0}\to
T_{cc;\bar{u}\bar{d} }^{+}\pi^{-}(K^{-}, D^{-}, D_{s}^{-})$, 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
$T_{bc}^{0}\to B^{\ast -}D^{+}$ and $T_{bc}^{0}\to \overline{B}^{\ast 0}D^{0}$,
and using these channels evaluate the full width of $T_{bc}^{0}$. Predictions
for $\Gamma_{ \mathrm{full}} =(3.98\pm 0.51)\times 10^{-10}~\mathrm{MeV}$ and
mean lifetime $\tau=1.65_{-0.18}^{+0.25}~\mathrm{ps}$ of $T_{bc}^{0}$ obtained
in the context of the first option, as well as the full width $\Gamma_{\mathrm{
full}}=(63.5\pm 8.9)~\mathrm{MeV}$ extracted in the second scenario may be
useful for experimental and theoretical exploration of double-heavy exotic
mesons.
