We present the first full-fledged study of the flavor-exotic isoscalar $T_{bb}^{-}\equiv bb\overline{u}\overline{d}$ tetraquark with spin and parity $J^P=1^{+}$. We report accurate solutions of the four-body problem in a quark model, characterizing the structure of the state as a function of the ratio $M_Q/m_q$ of the heavy to light quark masses. For such a standard constituent model, $T_{bb}^{-}$ lies approximately 150 MeV below the strong decay threshold $B^{-}{\overline{B^{⁎}}}^0$ and 105 MeV below the electromagnetic decay threshold $B^{-}{\overline{B}}^0\gamma$. We evaluate the lifetime of $T_{bb}^{-}$, identifying the promising decay modes where the tetraquark might be looked for in future experiments. Its total decay width is $\Gamma \approx 87\times {10}^{-15}$ GeV and therefore its lifetime τ≈ 7.6 ps. The promising final states are ${B^{⁎}}^{-}{D^{⁎}}^{+}{\ell }^{-}{\overline{\nu }}_{\ell }$ and ${\overline{B^{⁎}}}^0{D^{⁎}}^0{\ell }^{-}{\overline{\nu }}_{\ell }$ among the semileptonic decays, and ${B^{⁎}}^{-}{D^{⁎}}^{+}{D_s^{⁎}}^{-}$, ${\overline{B^{⁎}}}^0{D^{⁎}}^0{D_s^{⁎}}^{-}$, and ${B^{⁎}}^{-}{D^{⁎}}^{+}{\rho }^{-}$ among the nonleptonic ones. The semileptonic decay to the isoscalar $J^P=0^{+}$ tetraquark $T_{bc}^0$ is also relevant but it is not found to be dominant. There is a broad consensus about the existence of this tetraquark, and its detection will validate our understanding of the low-energy realizations of Quantum Chromodynamics (QCD) in the multiquark sector.