  The observed spatial distribution of $\gamma-$ray bursts indicates that they
probably originate at cosmological distances. At this distance scale their
variability timescale and flux above MeV imply an initial optical-depth to pair
production $> 10^{10}$. This appears to be in conflict with their highly
non-thermal spectra. We show that this difficulty can be removed if axion
bursts from supernovae are converted to $\gamma-$rays over cosmological
distances. Nonthermal bursts with the relevant flux, duration, variability and
spectra are obtained just for the range of axion masses of $10^{-5}-10^{-4}{\rm
eV}$ that accounts for the cold dark matter in the universe. The observed rate
of $\gamma-$ray bursts implies that axions should be converted efficiently to
photons in only one out of $\sim 10^{4}$ supernovae.
