Conventional theory predicts that ultrahigh lattice thermal conductivity can only occur incrystals composed of strongly-bonded light elements, and that it is limited by anharmonic three-phonon processes. We report experimental evidence that is a departure from these long-held criteria. We measured a local thermal conductivity exceeding 1000 W m-1 K-1 and an average bulk value close to 1000 W m-1 K-1 in bulk boron arsenide (BAs) crystals at room temperature. The high values are consistent with a proposal for phonon band engineering and can only be explained with higher order phonon processes in the compound semiconductor that contains both light boron and heavy arsenic elements.