We demonstrate reproducible voltage induced non-volatile switching of the magnetization in an epitaxial thin Fe81Ga19 film. Switching is induced at room temperature and without the aid of an external magnetic field. This is achieved by the modification of the magnetic anisotropy by mechanical strain induced by a piezoelectric transducer attached to the layer. Epitaxial Fe81Ga19 is shown to possess the favourable combination of cubic magnetic anisotropy and large magnetostriction necessary to achieve this functionality with experimentally accessible levels of strain. The switching of the magnetization proceeds by the motion of magnetic domain walls, also controlled by the voltage induced strain.