We have calculated the thermal conductivities (κ) of cubic III-V boron compounds using a predictive first principles approach. Boron arsenide is found to have a remarkable room temperature κ over 2000 W m(-1) K(-1); this is comparable to those in diamond and graphite, which are the highest bulk values known. We trace this behavior in boron arsenide to an interplay of certain basic vibrational properties that lie outside of the conventional guidelines in searching for high κ materials, and to relatively weak phonon-isotope scattering. We also find that cubic boron nitride and boron antimonide will have high κ with isotopic purification. This work provides new insight into the nature of thermal transport at a quantitative level and predicts a new ultrahigh κ material of potential interest for passive cooling applications.