The radio galaxy Fornax A is well known for its giant radio lobes which extend almost a degree (~300 kpc) across the sky. Fomalont et al. have shown that these lobes are linearly polarized on the largest scales, although the western lobe is highly depolarized in discrete, resolved regions. The depolarized regions indicate the presence of ionized gas (Faraday screen) along the line of sight to the western lobe. We have now detected the warm gas at H alpha using a Fabry-Perot interferometer in a way that allows us to reach very low surface brightness levels (<10-19 erg cm-2 s-1 arcsec-2). The radial velocity of the ionized gas (1610 km s-1) places it in the Fornax cluster at a projected radius of 1.0 Mpc. The gas is spread over a region 10 kpc in diameter and has an ionized mass of 6 x 107 f-0.5 M⊙ (where f is the volume filling factor). For the inferred column density of electrons (~2.6 x 1020 f-0.5 cm-2) and in light of the H I upper limits (<4 x 1019 cm-2), the cloud has to be mostly ionized. Possible sources of ionization are large-scale shocks or a putative hot component which confines the radio lobes. The high line-of-sight magnetic field strength (B|| ~ 0.3--1.3 mu G) deduced from the rotation measure (~20 rad m-2) and energy equipartition suggests that the cloud may have undergone significant turbulent heating from secondary shocks. The predicted bounds on the fraction of neutral material (10-6--10-1) indicates that this may be the first example of a spatially resolved, extragalactic ionized cloud. Such objects are thought to be responsible for some of the Ly alpha absorption lines observed in the spectra of quasars.