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Improving the thermal management of small-scale devices requires developing materials with high thermal conductivities. The semiconductor boron arsenide (BAs) is an attractive target because of ab initio calculation indicating that single crystals have an ultrahigh thermal conductivity. We synthesized BAs single crystals without detectable defects and measured a room-temperature thermal conductivity of 1300 watts per meter-kelvin. Our spectroscopy study, in conjunction with atomistic theory, reveals that the distinctive band structure of BAs allows for very long phonon mean free paths and strong high-order anharmonicity through the four-phonon process. The single-crystal BAs has better thermal conductivity than other metals and semiconductors. Our study establishes BAs as a benchmark material for thermal management applications and exemplifies the power of combining experiments and ab initio theory in new materials discovery.