Massive outflows driven by active galactic nuclei are widely recognized to have a key role in the evolution of galaxies, by heating the ambient gas, expelling it from the nuclear regions, and thereby affecting the star-formation histories of the galaxy bulges. It has been proposed that the powerful jets of relativistic particles (such as electrons) launched by some active nuclei can both accelerate and heat the molecular gas, which often dominates the mass budgets of the outflows. Clear evidence for this mechanism, in the form of detailed associations between the molecular gas kinematics and features in the radio-emitting jets, has however been lacking. Here we report that the warm molecular hydrogen gas in the western radio lobe of the Seyfert galaxy IC 5063 is moving at high velocities--up to about 600 kilometres per second--relative to the galaxy disk. This suggests that the molecules have been accelerated by fast shocks driven into the interstellar medium by the expanding radio jets. These results demonstrate the general feasibility of accelerating molecular outflows in fast shocks driven by active nuclei.