  We present the results of a calculation of the rate for the annihilation of
an $e^+e^-$ pair into a jet containing an energetic photon and a single further
jet at order $\alpha \alpha_{s}$. By comparing these results with the existing
data on the photon + 1 jet rate from the ALEPH Collaboration, we make a first
determination of the process independent quark-to-photon fragmentation function
$D_{q \to \gamma}(z,\mu_{F})$ at order $\alpha \alpha_{s}$. We also compare our
prediction for the inclusive photon energy spectrum with the recent OPAL
measurement.
