We generate coherent ultraviolet radiation at 313 nm as the third harmonic of an external-cavity diode laser. We use this radiation for laser cooling of trapped beryllium atomic ions and sympathetic cooling of co-trapped beryllium-hydride molecular ions. An LBO crystal in an enhancement cavity generates the second harmonic, and a BBO crystal in a doubly resonant enhancement cavity mixes this second harmonic with the fundamental to produce the third harmonic. Each enhancement cavity is preceded by a tapered amplifier to increase the fundamental light. The 36-mW output power of this all-semiconductor-gain system will enable quantum control of the beryllium ions' motion.