The potentials between two $B$ mesons are computed in the heavy-quark limit using quenched lattice QCD at $m_{\pi }\sim 400\mathrm{MeV}$. Nonzero central potentials are clearly evident in all four spin-isospin channels, $(I,s_l)=(0,0)$, (0, 1), (1, 0), (1, 1), where $s_l$ is the total spin of the light degrees of freedom. At short distance, we find repulsion in the $I\ne s_l$ channels and attraction in the $I=s_l$ channels. Linear combinations of these potentials that have well-defined spin and isospin in the $t$-channel are found, in three of the four cases, to have substantially smaller uncertainties than the potentials defined with the $s$-channel $(I,s_l)$, and allow quenching artifacts from single hairpin exchange to be isolated. The $B{B}^{*}\pi$ coupling extracted from the long-distance behavior of the finite-volume $t$-channel potential is found to be consistent with quenched calculations of the matrix element of the isovector axial-current. The tensor potentials in both of the $s_l=1$ channels are found to be consistent with zero within calculational uncertainties.

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