Radiative Corrections to the Photon + 1 Jet Rate at LEP

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 $\alpha \alpha_{s}$.