  The potentials between two B-mesons are computed in the heavy-quark limit
using quenched lattice QCD at $m_\pi\sim 400~{\rm MeV}$. Non-zero 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 BB*\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.
