Quantum systems in confined geometries are host to novel physical phenomena. Examples include quantum Hall systems in semiconductors(1) and Dirac electrons in graphene(2). Interest in such systems has also been intensified by the recent discovery of a large enhancement in photoluminescence quantum efficiency(3-7) and a potential route to valleytronics(6-8) in atomically thin layers of transition metal dichalcogenides, MX2 (M = Mo, W; X = S, Se, Te), which are closely related to the indirect-to-direct bandgap transition in monolayers(9-12). Here, we report the first direct observation of the transition from indirect to direct bandgap in monolayer samples by using angle-resolved photoemission spectroscopy on high-quality thin films of MoSe2 with variable thickness, grown by molecular beam epitaxy. The band structure measured experimentally indicates a stronger tendency of monolayer MoSe2 towards a direct bandgap, as well as a larger gap size, than theoretically predicted. Moreover, our finding of a significant spin-splitting of similar to 180 meV at the valence band maximum of a monolayer MoSe2 film could expand its possible application to spintronic devices.