  We extend HPQCD's earlier $n_f=4$ lattice-QCD analysis of the ratio of
$\overline{\mathrm{MSB}}$ masses of the $b$ and $c$ quark to include results
from finer lattices (down to 0.03fm) and a new calculation of QED contributions
to the mass ratio. We find that
$\overline{m}_b(\mu)/\overline{m}_c(\mu)=4.586(12)$ at renormalization scale
$\mu=3$\,GeV. This result is nonperturbative. Combining it with HPQCD's recent
lattice QCD$+$QED determination of $\overline{m}_c(3\mathrm{GeV})$ gives a new
value for the $b$-quark mass: $\overline{m}_b(3\mathrm{GeV}) = 4.513(26)$GeV.
The $b$-mass corresponds to $\overline{m}_b(\overline{m}_b, n_f=5) =
4.202(21)$GeV. These results are the first based on simulations that include
QED.
