There is much recent interest in exploiting the spin of conduction electrons in semiconductor heterostructures together with their charge to realize new device concepts. Electrical currents are usually generated by electric or magnetic fields, or by gradients of, for example, carrier concentration or temperature. The electron spin in a spin-polarized electron gas can, in principle, also drive an electrical current, even at room temperature, if some general symmetry requirements are met. Here we demonstrate such a 'spin-galvanic' effect in semiconductor heterostructures, induced by a non-equilibrium, but uniform population of electron spins. The microscopic origin for this effect is that the two electronic sub-bands for spin-up and spin-down electrons are shifted in momentum space and, although the electron distribution in each sub-band is symmetric, there is an inherent asymmetry in the spin-flip scattering events between the two sub-bands. The resulting current flow has been detected by applying a magnetic field to rotate an optically oriented non-equilibrium spin polarization in the direction of the sample plane. In contrast to previous experiments, where spin-polarized currents were driven by electric fields in semiconductor, we have here the complementary situation where electron spins drive a current without the need of an external electric field.