The mass and width of the tensor tetraquark $T=bb\bar{c}\bar{c}$ with spin-parity $J^P=2^{+}$ are calculated in the context of the QCD sum rule method. The tetraquark T is modeled as a diquark-antidiquark state built of components $b^{T}C{\gamma }_{\mu }b$ and $\bar{c}{\gamma }_{\nu }C{\bar{c}}^T$ with C being the charge conjugation matrix. The mass $m=(12.795\pm 0.095)\mathrm{GeV}$ of the exotic tensor meson T is found by means of the two-point sum rule approach. Its full width Γ is evaluated by considering processes $T\to B_c^{-}{B}_c^{-}$, $B_c^{-}{B}_c^{⁎-}$, and $B_c^{⁎-}{B}_c^{⁎-}$. Partial widths of these decays are computed by means of the three-point sum rule approach which is used to determine the strong couplings at relevant tetraquark-meson-meson vertices. Predictions obtained for the width ${\Gamma }_T={55.5}_{-9.9}^{+10.6}\mathrm{MeV}$, as well as the mass of the tetraquark T can be useful in investigations of fully heavy four-quark mesons.