  We perform a lattice QCD calculation of the hadronic light-by-light
contribution to $(g-2)_\mu$ at the SU(3) flavor-symmetric point
$m_\pi=m_K\simeq 420\,$MeV. The representation used is based on
coordinate-space perturbation theory, with all QED elements of the relevant
Feynman diagrams implemented in continuum, infinite Euclidean space. As a
consequence, the effect of using finite lattices to evaluate the QCD four-point
function of the electromagnetic current is exponentially suppressed. Thanks to
the SU(3)-flavor symmetry, only two topologies of diagrams contribute, the
fully connected and the leading disconnected. We show the equivalence in the
continuum limit of two methods of computing the connected contribution, and
introduce a sparse-grid technique for computing the disconnected contribution.
Thanks to our previous calculation of the pion transition form factor, we are
able to correct for the residual finite-size effects and extend the tail of the
integrand. We test our understanding of finite-size effects by using gauge
ensembles differing only by their volume. After a continuum extrapolation based
on four lattice spacings, we obtain $a_\mu^{\rm hlbl} = (65.4\pm 4.9 \pm
6.6)\times 10^{-11}$, where the first error results from the uncertainties on
the individual gauge ensembles and the second is the systematic error of the
continuum extrapolation. Finally, we estimate how this value will change as the
light-quark masses are lowered to their physical values.
