Unidirectional electric current flow, such as that found in a diode, is essential for modern electronics. It usually occurs at asymmetric interfaces such as p-n junctions or metal/semiconductor interfaces with Schottky barriers. We report on a diode effect associated with the direction of bulk electric polarization in BiFeO3: a ferroelectric with a small optical gap edge of approximately 2.2 electron volts. We found that bulk electric conduction in ferroelectric monodomain BiFeO3 crystals is highly nonlinear and unidirectional. This diode effect switches its direction when the electric polarization is flipped by an external voltage. A substantial visible-light photovoltaic effect is observed in BiFeO3 diode structures. These results should improve understanding of charge conduction mechanisms in leaky ferroelectrics and advance the design of switchable devices combining ferroelectric, electronic, and optical functionalities.