We present high-spatial resolution 21-cm HI VLA observations of the radio galaxy 3C 305 ( z = 0.041). These new high-resolution data show that the similar to 1000 km s(-1) broad H. absorption, earlier detected in low-resolution WSRT observations, is occurring against the bright, eastern radio lobe, about 1.6 kpc from the nucleus. We use new optical spectra taken with the WHT to make a detailed comparison of the kinematics of the neutral hydrogen with that of the ionised gas. The striking similarity between the complex kinematics of the two gas phases suggests that both the ionised gas and the neutral gas are part of the same outflow. Earlier studies of the ionised gas had already found evidence for a strong interaction between the radio jet and the interstellar medium at the location of the eastern radio lobe. Our results show that the fast outflow produced by this interaction also contains a component of neutral atomic hydrogen. The most likely interpretation is that the radio jet ionises the ISM and accelerates it to the high outflow velocities observed. Our observations demonstrate that, following this strong jet-cloud interaction, not all gas clouds are destroyed and that part of the gas can cool and become neutral. The mass outflow rate measured in 3C 305 is comparable, although at the lower end of the distribution, to that found in Ultra-Luminous IR galaxies. This suggests that AGN-driven outflows, and in particular jet-driven outflows, can have a similar impact on the evolution of a galaxy as starburst-driven superwinds.