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.
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