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Hexagonal boron nitride (hBN) is a wide-gap material that has attracted significant attention as an ideal dielectric substrate for 2D crystal heterostructures. We report here the first observation of in-plane charge transport in large-area monolayer hBN, grown by chemical vapor deposition. The quadratic scaling of current with voltage at high bias corresponds to a space-charge limited conduction mechanism, with a room-temperature mobility reaching up to 0.01 cm²/(V s) at electric fields up to 100 kV/cm in the absence of dielectric breakdown. The observation of in-plane charge transport highlights the semiconducting nature of monolayer hBN, and identifies hBN as a wide-gap 2D crystal capable of supporting charge transport at high field. Future exploration of charge transport in hBN is motivated by the fundamental study of UV optoelectronics and the massive Dirac fermion spectrum of hBN.