  The transparent semiconductor In$_{2}$O$_{3}$ is a technologically important
material. It combines optical transparency in the visible frequency range and
sizeable electric conductivity. We present a study of thermal conductivity of
In$_{2}$O$_{3}$ crystals and find that around 20 K, it peaks to a value as high
as 5,000 WK$^{-1}$m$^{-1}$, comparable to the peak thermal conductivity in
silicon and exceeded only by a handful of insulators. The amplitude of the peak
drastically decreases in presence of a type of disorder, which does not simply
correlate with the density of mobile electrons. Annealing enhances the ceiling
of the phonon mean free path. Samples with the highest thermal conductivity are
those annealed in the presence of hydrogen. Above 100 K, thermal conductivity
becomes sample independent. In this intrinsic regime, dominated by
phonon-phonon scattering, the magnitude of thermal diffusivity, $D$ becomes
comparable to many other oxides, and its temperature dependence evolves towards
$T^{-1}$. The ratio of $D$ to the square of sound velocity yields a scattering
time which obeys the expected scaling with the Planckian time.
