Conventional computer electronics creates a dichotomy between how information is processed and how it is stored. Silicon chips process information by controlling the flow of charge through a network of logic gates. This information is then stored, most commonly, by encoding it in the orientation of magnetic domains of a computer hard disk. The key obstacle to a more intimate integration of magnetic materials into devices and circuit processing information is a lack of efficient means to control their magnetization. This is usually achieved with an external magnetic field or by the injection of spin-polarized currents. The latter can be significantly enhanced in materials whose ferromagnetic properties are mediated by charge carriers. Among these materials, conductors lacking spatial inversion symmetry couple charge currents to spin by intrinsic spin-orbit (SO) interactions, inducing nonequilibrium spin polarization tunable by local electric fields. Here we show that magnetization of a ferromagnet can be reversibly manipulated by the SO-induced polarization of carrier spins generated by unpolarized currents. Specifically, we demonstrate domain rotation and hysteretic switching of magnetization between two orthogonal easy axes in a model ferromagnetic semiconductor.