  Parameters of the heavy four-quark scalar meson $T_{\mathrm{2bc}}=bc
\overline{b}\overline{c}$ are calculated by means of the sum rule method. This
structure is considered as a diquark-antidiquark state built of scalar diquark
and antidiquark components. The mass and current coupling of $T_{
\mathrm{2bc}}$ are evaluated in the context of the two-point sum rule approach.
The full width of this tetraquark is estimated by taking into account two types
of its possible strong decay channels. First class includes dissociation of
$T_{\mathrm{2bc}}$ to mesons $\eta_c\eta_{b}$, $ B_{c}^{+}B_{c}^{-}$,
$B_{c}^{\ast +}B_{c}^{\ast -}$ and $ B_{c}^{+}(1^3P_{0})B_{c}^{\ast-}$. Another
type of processes are generated by annihilations $\overline{b}b \to
\overline{q}q$ of constituent $b$-quarks which produces the final-state charmed
meson pairs $D^{+}D^{-}$, $D^{0} \overline{D}^{0}$, $D^{*+}D^{*-}$, and
$D^{*0}\overline{D}^{*0}$. Partial width all of these decays are found using
the three-point sum rule method which is required to calculate strong couplings
at corresponding meson-meson-tetraquark vertices. Predictions obtained for the
mass $m=(12697 \pm 90)~\mathrm{MeV}$ and width $\Gamma[T_{\mathrm{2bc}}]=(142.4
\pm 16.9)~ \mathrm{MeV}$ of the state $T_{\mathrm{2bc}}$ are compared with
alternative results, and are useful for further experimental investigations of
fully heavy resonances.
