  We improve the precision of the topological susceptibility of QCD, and
therefore of the QCD axion mass, by including $O(\alpha_{\rm em})$ and NNLO
corrections in the chiral expansion, which amount to 0.65(21)% and -0.71(29)%
respectively. Both corrections are one order of magnitude smaller than the
known NLO ones, confirming the very good convergence of the chiral expansion
and its reliability. Using the latest estimates for the light quark masses the
current uncertainty is dominated by the one of the low-energy constant
$\ell_7$. When combined with possible improvements on the light quark mass
ratio and $\ell_7$ from lattice QCD, our computation could allow to determine
the QCD axion mass with per-mille accuracy.
