  We report on a calculation of the $B^*B\pi$ coupling in lattice QCD. The
strong matrix element $\langle B \pi | B^*\rangle$ is directly related to the
leading order low-energy constant in heavy meson chiral perturbation theory
(HM$\chi$PT) for $B$-mesons. We carry out our calculation directly at the
$b$-quark mass using a non-perturbatively tuned clover action that controls
discretisation effects of order $|\vec{p}a|$ and $(ma)^n$ for all $n$. Our
analysis is performed on RBC/UKQCD gauge configurations using domain wall
fermions and the Iwasaki gauge action at two lattice spacings of
$a^{-1}=1.73(3)$ GeV, $a^{-1}=2.28(3)$ GeV, and unitary pion masses down to 290
MeV. We achieve good statistical precision and control all systematic
uncertainties, giving a final result for the HM$\chi$PT coupling $g_b =
0.569(48)_{stat}(59)_{sys}$ in the continuum and at the physical light-quark
masses. This is the first calculation performed directly at the physical
$b$-quark mass and lies in the region one would expect from carrying out an
interpolation between previous results at the charm mass and at the static
point.
