Stacking two semiconducting transition metal dichalcogenide (MX2) monolayers to form a heterobilayer creates a new variety of semiconductor junction with unique optoelectronic features, such as hosting long-lived dipolar interlayer excitons. Despite many optical, transport, and theoretical studies, there have been few direct electronic structure measurements of these junctions. Here, we apply angle-resolved photoemission spectroscopy with micron-scale spatial resolution (µARPES) to determine the band alignments in MoSe2/WSe2 heterobilayers, using in-situ electrostatic gating to electron-dope and thus probe the conduction band edges. By comparing spectra from heterobilayers with opposite stacking orders, that is, with either MoSe2 or WSe2 on top, we confirm that the band alignment is type II, with the valence band maximum in the WSe2 and the conduction band minimum in the MoSe2. The overall band gap is EG = 1.43 ± 0.03 eV, and to within experimental uncertainty it is unaffected by electron doping. However, the offset between the WSe2 and MoSe2 valence bands clearly decreases with increasing electron doping, implying band renormalisation only in the MoSe2, the layer in which the electrons accumulate. In contrast, µARPES spectra from a WS2/MoSe2 heterobilayer indicate type I band alignment, with both band edges in the MoSe2. These insights into the doping-dependent band alignments and gaps of MX2 heterobilayers will be useful for properly understanding and ultimately utilizing their optoelectronic properties.