  The QCD axion offers a natural resolution to the strong CP problem and
provides a compelling dark matter candidate. If the QCD axion constitutes all
the dark matter, the simplest models pick out a narrow range of masses around
$100\,\mu{\rm eV}$. We point out a natural production mechanism for QCD axion
dark matter at masses up to existing astrophysical bounds (${\scr O}(20 \,
\mathrm{meV})$ for the most minimal models and ${\scr O}(1 \, \mathrm{eV})$ for
nucleophobic models). If the QCD axion mixes with a sterile axion, the relative
temperature dependence of their potentials can lead to an avoided level
crossing of their mass eigenstates. This leads to a near-total transfer of
energy density from the sterile axion to the QCD axion, resulting in a
late-time QCD axion abundance sufficient to make up all of present-day dark
matter. Our result provides additional theoretical motivation for several
direct detection experiments that will probe this part of parameter space in
the near future.
