  Recent improvements to atomic energy-level data allow, for the first time,
accurate predictions to be made for the Fe III line emission strengths in the
spectra of luminous, $L_\text{bol}=10^{46}-10^{48}$ erg/s, Active Galactic
Nuclei. The Fe III emitting gas must be primarily photoionized, consistent with
observations of line reverberation. We use CLOUDY models exploring a wide range
of parameter space, together with 26,500 rest-frame ultraviolet spectra from
the Sloan Digital Sky Survey, to constrain the physical conditions of the line
emitting gas. The observed Fe III emission is best accounted for by dense
($n_H=10^{14}$ cm$^{-3}$) gas which is microturbulent, leading to smaller line
optical depths and fluorescent excitation. Such high density gas appears to be
present in the central regions of the majority of luminous quasars. Using our
favoured model, we present theoretical predictions for the relative strengths
of the Fe III UV34 $\lambda\lambda$1895,1914,1926 multiplet. This multiplet is
blended with the Si III] $\lambda$1892 and C III] $\lambda$1909 emission lines
and an accurate subtraction of UV34 is essential when using these lines to
infer information about the physics of the broad line region in quasars.
