  The mass and width of the tensor tetraquark $T=bb\overline{c}\overline{c}$
with spin-parity $J^{\mathrm{P}}=2^{+}$ 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 $\overline{c}\gamma _{\nu
}C\overline{c}^{T}$ 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 $\Gamma$ is evaluated
by considering processes $T \to B_{c}^{-}B_{c}^{-}$, $ B_{c}^{-}B_{c}^{\ast
-}$, and $B_{c}^{\ast -}B_{c}^{\ast -}$. 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 $\Gamma
_{\mathrm{T}}=55.5_{-9.9}^{+10.6}~\mathrm{MeV}$, as well as the mass of the
tetraquark $T $ can be useful in investigations of fully heavy four-quark
mesons.
