As they combine decent mobilities with extremely long carrier lifetimes, organic-inorganic perovskites have opened a whole new field in opto-electronics. Measurements of their underlying electronic structure, however, are still lacking. Using angle-resolved photoelectron spectroscopy, we measure the valence band dispersion of single-crystal CH$_3$NH$_3$PbBr$_3$. The dispersion of the highest energy band is extracted applying a modified leading edge method, which accounts for the particular density of states of organic-inorganic perovskites. The surface Brillouin zone is consistent with bulk-terminated surfaces both in the low-temperature orthorhombic and the high-temperature cubic phase. In the low-temperature phase, we find a ring-shaped valence band maximum with a radius of 0.043 {AA}$^{-1}$, centered around a 0.16 eV deep local minimum in the dispersion of the valence band at the high-symmetry point. Intense circular dichroism is observed. This dispersion is the result of strong spin-orbit coupling. Spin-orbit coupling is also present in the room-temperature phase. The coupling strength is one of the largest reported so far.
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