In the past year, the HiRes and Auger collaborations have reported the discovery of a high-energy cut-off in the ultra-high-energy cosmic ray spectrum, and an apparent clustering of the highest energy events towards nearby active galactic nuclei. Consensus is building that such ~10¹⁹-10²⁰ eV particles are accelerated within the radio-bright lobes of these sources, but it is not yet clear how this actually happens. In this paper, we report (to our knowledge) the first treatment of stochastic particle acceleration in such environments from first principles, showing that energies ~10²⁰ eV are reached in ~10⁶ yr for protons. However, our findings reopen the question regarding whether the high-energy cut-off is due solely to propagation effects, or whether it does in fact represent the maximum energy permitted by the acceleration process itself.