  In the past year, the HiRes and Auger collaborations have reported the
discovery of a high-energy cutoff in the ultra-high energy cosmic-ray (UHECR)
spectrum, and an apparent clustering of the highest energy events towards
nearby active galactic nuclei (AGNs). Consensus is building that such $\sim
10^{19}$--$10^{20}$ 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
$\sim 10^{20}$ eV are reached in $\sim 10^6$ years for protons. However, our
findings reopen the question regarding whether the high-energy cutoff is due
solely to propagation effects, or whether it does in fact represent the maximum
energy permitted by the acceleration process itself.
