  We present a complete calculation of the photon +~1 jet rate in $e^+e^-$
annihilation up to ${\cal O}(\alpha \alpha_{s})$. Although formally of
next-to-leading order in perturbation theory, this calculation contains several
ingredients appropriate to a next-to-next-to-leading order calculation of jet
observables. In particular, we describe a generalization of the commonly used
phase space slicing method to isolate the singularities present when more than
one particle is unresolved. Within this approach, we analytically evaluate the
singularities associated with the following double unresolved regions; triple
collinear, soft/collinear and double single collinear configurations as well as
those from the collinear limit of virtual graphs. By comparing the results of
our calculation with the existing data on the photon +~1 jet rate from the
ALEPH Collaboration at CERN, we make a next-to-leading order determination of
the process-independent non-perturbative quark-to-photon fragmentation function
$D_{q \to \gamma}(z,\mu_{F})$ at ${\cal O}(\alpha \alpha_{s})$. As a first
application of this measurement allied with our improved perturbative
calculation, we determine the dependence of the isolated photon +~1 jet cross
section in a democratic clustering approach on the jet resolution parameter
$\ycut$ at next-to-leading order. The next-to-leading order corrections to this
observable are moderate but improve the agreement between theoretical
prediction and experimental data.
