A new mechanism for the formation of binary systems through the fragmentation of a protostellar disc is presented. This process requires a strongly rotationally unstable (to both m = 2 and m = 1 modes), central accreting object surrounded by a rotationally supported disc of gas which is subject to continuing infall. The central instability becomes off-centre due to the growth of the m = 1 mode. The interaction of the spiral arms thus driven into the disc, with the continued infall, gives rise to the formation of a self-gravitating secondary body in orbit around the first. For cores that form without discs, the core evolves as in past studies of rotational instabilities. In addition to the m = 2 bar mode, an m = 1 mode develops, allowing the protostar to convert some of its spin angular momentum into orbital angular momentum. Furthermore, the gravitational torques from the bar directly account for the outward transfer of angular momentum. For cores that form with surrounding discs and with continuing infall, the spiral arms from the core move through the disc gathering matter together. The combination of the m = 2 mode and the displacement of the core from the m = 1 mode allows the spiral arms to gather sufficient matter together from the disc and the continuing infall to become gravitationally unstable. This matter then collapses to form a secondary. In cases where there is no continuing infall on to the disc, the spiral arms (from the m = 1 and m = 2 modes) are not able to gather sufficient material to form a Jeans unstable condensation.
