  It has been pointed out in the literature that in the presence of an external
magnetic field the axion mass receives an electromagnetic contribution. We show
that if a magnetic field with energy density larger than ~10^{-8} times the
energy density of the Universe existed at temperatures of a few GeV, that
contribution would be dominant and consequently the cosmic evolution of the
axion field would change substantially. In particular, the expected axion relic
abundance would be lowered, allowing a small relaxation of the present
cosmological bound on the Peccei-Quinn constant.
