We present the optical discovery and subarcsecond optical and X-ray localization of the afterglow of the short GRB 120804A, as well as optical, near-IR, and radio detections of its host galaxy. X-ray observations with Swift/XRT, Chandra, and XMM-Newton extending to δt ≈ 19 days reveal a single power-law decline. The optical afterglow is faint, and comparison to the X-ray flux indicates that GRB 120804A is "dark," with a rest-frame extinction of A host V ≈ 2.5 mag (at z = 1.3). The intrinsic neutral hydrogen column density inferred from the X-ray spectrum, N H, int(z = 1.3) ≈ 2 × 1022 cm–2, is commensurate with the large extinction. The host galaxy exhibits red optical/near-IR colors. Equally important, JVLA observations at ≈0.9-11 days reveal a constant flux density of F ν(5.8 GHz) = 35 ± 4 μJy and an optically thin spectrum, unprecedented for GRB afterglows, but suggestive instead of emission from the host galaxy. The optical/near-IR and radio fluxes are well fit with the scaled spectral energy distribution of the local ultraluminous infrared galaxy (ULIRG) Arp 220 at z ≈ 1.3, with a resulting star formation rate of x ≈ 300 M ☉ yr–1. The inferred extinction and small projected offset (2.2 ± 1.2 kpc) are also consistent with the ULIRG scenario, as is the presence of a companion galaxy at the same redshift and with a separation of about 11 kpc. The limits on radio afterglow emission, in conjunction with the observed X-ray and optical emission, require a circumburst density of n ~ 10–3 cm–3, an isotropic-equivalent energy scale of E γ, iso ≈ E K, iso ≈ 7 × 1051 erg, and a jet opening angle of θ j gsim 11°. The expected fraction of luminous infrared galaxies in the short GRB host sample is ~0.01 and ~0.25 (for pure stellar mass and star formation weighting, respectively). Thus, the observed fraction of two events in about 25 hosts (GRBs 120804A and 100206A) appears to support our previous conclusion that short GRBs track both stellar mass and star formation activity.