  Long carrier lifetimes and diffusion lengths form the basis for the
successful application of the organic-inorganic perovskite
(CH$_3$NH$_3$)PbI$_3$ in solar cells and lasers. The mechanism behind the long
carrier lifetimes is still not completely understood. Spin-split bands and a
resulting indirect band gap have been proposed by theory. Using near band-gap
left-handed and right-handed circularly polarized light we induce photocurrents
of opposite directions in a single-crystal (CH$_3$NH$_3$)PbI$_3$ device at low
temperature ($4~\mathrm{K}$). The phenomenom is known as the circular
photogalvanic effect and gives direct evidence for phototransport in spin-split
bands. Simultaneous photoluminecence measurements show that the onset of the
photocurrent is below the optical band gap. The results prove that an indirect
band gap exists in (CH$_3$NH$_3$)PbI$_3$ with broken inversion symmetry as a
result of spin-splittings in the band structure. This information is essential
for understanding the photophysical properties of organic-inorganic perovskites
and finding lead-free alternatives. Furthermore, the optically driven spin
currents in (CH$_3$NH$_3$)PbI$_3$ make it a candidate material for spintronics
applications.
