Parameters of the heavy four-quark scalar meson $T_{\mathrm{bc}\bar{b}\bar{c}}$ with content $bc\bar{b}\bar{c}$ 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 $T_{\mathrm{bc}\bar{b}\bar{c}}$ 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 $T_{\mathrm{bc}\bar{b}\bar{c}}$ to mesons ${\eta }_c{\eta }_b$, $B_c^{+}{B}_c^{-}$, $B_c^{⁎+}{B}_c^{⁎-}$ and $B_c^{+}\left(1^3{P}_0\right)B_c^{⁎-}$. Another type of processes are generated by annihilations $\bar{b}b\to \bar{q}q$ of constituent b-quarks which produces the final-state charmed meson pairs $D^{+}{D}^{-}$, $D^0{\bar{D}}^0$, $D^{⁎+}{D}^{⁎-}$, and $D^{⁎0}{\bar{D}}^{⁎0}$. 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 $\Gamma \left[T_{\mathrm{bc}\bar{b}\bar{c}}\right]=(142.4\pm 16.9)\mathrm{MeV}$ of this state are compared with alternative results, and are useful for further experimental investigations of fully heavy resonances.