We use high-resolution (0.5 arcsec) CO(2−1) observations performed with the Atacama Large Millimetre/submillimetre Array to trace the kinematics of the molecular gas in the Seyfert 2 galaxy IC 5063. The data reveal that the kinematics of the gas is very complex. A fast outflow of molecular gas extends along the entire radio jet (∼1 kpc), with the highest outflow velocities about 0.5 kpc from the nucleus, at the location of the brighter hot spot in the western lobe. The ALMA data show that a massive, fast outflow with velocities up to 650 km s −1 of cold molecular gas is present, in addition to the outflow detected earlier in warm H 2 , H i and ionized gas. All phases of the gas outflow show similar kinematics. IC 5063 appears to be one of the best examples of the multi-phase nature of AGN-driven outflows. Both the central AGN and the radio jet could energetically drive the outflow, however, the characteristics of the outflowing gas point to the radio jet being the main driver. This is an important result because IC 5063, although one of the most powerful Seyfert galaxies, is a relatively weak radio source (P 1.4 GHz = 3 × 10 23 W Hz −1 ). All the observed characteristics can be described by a scenario of a radio plasma jet expanding into a clumpy medium, interacting directly with the clouds and inflating a cocoon that drives a lateral outflow into the interstellar medium. This model is consistent with results obtained by recent simulations. A stronger, direct interaction between the jet and a gas cloud is present at the location of the brighter western lobe. This interaction may also be responsible for the asymmetry in the radio brightness of the two lobes. Even assuming the most conservative values for the conversion factor CO-to-H 2 , we find that the mass of the outflowing gas is between 1.9 and 4.8 × 10 7 M , of which between 0.5 and 1.3 × 10 7 M is associated with the fast outflow at the location of the western lobe. These amounts are much larger than those of the outflow of warm gas (molecular and ionized) and somewhat larger than of the H i outflow. This suggests that most of the observed cold molecular outflow is due to fast cooling after being shocked. This gas is the end product of the cooling process, although some of it could be the result of only partly dissociated clouds. Our CO observations demonstrate that fast outflows of substantial masses of molecular gas can be driven by relativistic jets, although in the case of IC 5063 the outflows are not fast enough to remove significant amounts of gas from the galaxy and the effects are limited to the central ∼0.5 kpc from the centre.