  We compute the one-loop QCD correction to the photon-quark-antiquark vertex
in an extremely strong magnetic background, i.e., one in which $\sqrt{eB}$ is
much larger than all other mass scales. We resort to the lowest-Landau level
approximation, and consider on shell fermions. We find that the total magnetic
moment is such that the anomalous magnetic moment contributes to the electric
part $\sim iE_3\sigma_3$, the other contributions to
$\boldsymbol{B}\cdot\boldsymbol{\sigma}$ and
$i\boldsymbol{E}\cdot\boldsymbol{\sigma}_\perp$ being completely suppressed. We
show results for the anomalous magnetic moment of quarks up and down, scaled by
their values in the vacuum, as a function of the external magnetic field,
considering dressed gluons and a renormalization scale $\mu_\mathrm{\tiny
\overline{ MS}}=\sqrt{|eB|}$.
