We examine the possibility that energy and angular momentum are removed magnetically from accretion discs, by field lines that leave the disc surface and extend to large distances. We illustrate this mechanism by solving the equations of magnetohydrodynamics, assuming infinite conductivity, for axially symmetric, self-similar, cold magnetospheric flow from a Keplerian accretion disc in which the field strength B scales with radius r as |$B\,\propto\,{r}^{-5/4}$.| We show that a centrifugally driven outflow of matter from the disc is possible, if the poloidal component of the magnetic field makes an angle of less than 60° with the disc surface. At large distances from the disc, the toroidal component of the magnetic field becomes important and collimates the outflow into a pair of anti-parallel jets moving perpendicular to the disc. Close to the disc, the flow is probably driven by gas pressure in a hot magnetically dominated corona. In this way, magnetic stresses can extract the angular momentum from a thin accretion disc and thus enable matter to be accreted, independently of the presence of viscosity. These jet solutions have the property that most of the power is concentrated within a central core, while most of the angular momentum and magnetic flux is carried near the jet walls. The relevance of this mechanism for the evolution of accretion discs around massive black holes in galactic nuclei and the production of jets in extragalactic radio sources is described.