We present the first full-fledged study of the flavor-exotic isoscalar $T_{bb}^-equiv b b bar u bar d$ 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 $B^-bar {B^*}^{0}$ and 105 MeV below the electromagnetic decay threshold $B^- bar B^0 gamma$. 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 $tau approx$ 7.6 ps. The promising final states are ${B^*}^{-}, {D^*}^{+} , ell^- , bar u_ell$ and $bar {B^*}^{0} , {D^*}^{0} , ell^- , bar u_ell $ among the semileptonic decays, and ${B^*}^{-} , {D^*}^{+} , {D_s^*}^-$, $bar {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.
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