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Heterojunctions between three-dimensional (3D) semiconductors with different bandgaps are the basis of modern light-emitting diodes, diode lasers and high-speed transistors. Creating analogous heterojunctions between different 2D semiconductors would enable band engineering within the 2D plane and open up new realms in materials science, device physics and engineering. Here we demonstrate that seamless high-quality in-plane heterojunctions can be grown between the 2D monolayer semiconductors MoSe2 and WSe2. The junctions, grown by lateral heteroepitaxy using physical vapour transport, are visible in an optical microscope and show enhanced photoluminescence. Atomically resolved transmission electron microscopy reveals that their structure is an undistorted honeycomb lattice in which substitution of one transition metal by another occurs across the interface. The growth of such lateral junctions will allow new device functionalities, such as in-plane transistors and diodes, to be integrated within a single atomically thin layer.