  If axions or other low-mass pseudoscalars couple to electrons (``fine
structure constant'' $\alpha_a$) they are emitted from red giant stars by the
Compton process $\gamma+e\to e+a$ and by bremsstrahlung $e+(Z,A)\to (Z,A)+e+a$.
We construct a simple analytic expression for the energy-loss rate for all
conditions relevant for a red giant and include axion losses in evolutionary
calculations from the main sequence to the helium flash. We find that
$\alpha_a\lapprox0.5\mn(-26)$ or $m_a\lapprox 9\,\meV/\cos^2\beta$ lest the red
giant core at helium ignition exceed its standard mass by more than
$0.025\,\MM_\odot$, in conflict with observational evidence. Our bound is the
most restrictive limit on $\alpha_a$, but it does not exclude the possibility
that axion emission contributes significantly to the cooling of ZZ~Ceti stars
such as G117--B15A for which the period decrease was recently measured.
