  We study semileptonic decays of the scalar tetraquark $Z_{bc;\overline{u}
\overline{d}}^{0}$ to final states $T_{bs;\overline{u}\overline{d}
}^{-}e^{+}\nu_{e}$ and $T_{bs;\overline{u}\overline{d}}^{-}\mu^{+}\nu_{\mu}$ ,
which run through the weak transitions $c\to se^{+}\nu_{e}$ and $c\to
s\mu^{+}\nu_{\mu}$, respectively. To this end, we calculate the mass and
coupling of the final-state scalar tetraquark $T_{bs;\overline{u}\overline{d}
}^{-} $ by means of the QCD two-point sum rule method: these spectroscopic
parameters are used in our following investigations. In calculations we take
into account the vacuum expectation values of the quark, gluon, and mixed
operators up to dimension ten. We use also three-point sum rules to evaluate
the weak form factors $G_{i}(q^2)$ ($i=1,~2$) that describe these decays. The
sum rule predictions for $G_{i}(q^2)$ are employed to construct fit functions
$F_{i}(q^2)$, which allow us to extrapolate the form factors to the whole
region of kinematically accessible $q^2$. These functions are required to get
partial widths of the semileptonic decays $\Gamma \left( Z_{bc}^{0}\rightarrow
Te^{+}\nu_{e}\right) $ and $\Gamma \left( Z_{bc}^{0}\rightarrow T\mu
^{+}\nu_{\mu }\right)$ by integrating corresponding differential rates. We
analyze also the two-body nonleptonic decays
$Z_{bc;\overline{u}\overline{d}}^{0} \to T_{bs;\overline{u}\overline{d
}}^{-}\pi^{+}$ and $Z_{bc;\overline{u}\overline{d}}^{0} \to T_{bs;\overline{u
}\overline{d}}^{-}K^{+}$, which are necessary to evaluate the full width of the
$Z_{bc;\overline{u}\overline{d}}^{0}$. The obtained results for $\Gamma_{
\mathrm{full}}=(3.18\pm 0.39)\times 10^{-11}~\mathrm{MeV}$ and mean lifetime
$20.7_{-2.3}^{+2.9}~\mathrm{ps}$ of the tetraquark $Z_{bc;\overline{u}
\overline{d}}^{0}$ can be used in experimental investigations of this exotic
state.
