We present the optical discovery and sub-arcsecond 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 to ~19 d 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_V~2.5 mag (at z~1.3). The intrinsic neutral hydrogen column density inferred from the X-ray spectrum, N_H~2x10^22 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 d reveal a constant 5.8 GHz flux density 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 ultra-luminous infrared galaxy (ULIRG) Arp 220 at z~1.3, with a resulting star formation rate of ~300 Msun/yr. 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 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 ~10^-3 cm^-3 an isotropic-equivalent energy scale of E_gamma,iso ~ E_K,iso ~ 7x10^51 erg, and a jet opening angle of >8 deg. The expected fraction of luminous infrared galaxies in the short GRB host sample is ~0.01-0.3 (for pure stellar mass and star formation weighting, respectively). Thus, the observed fraction of 2 events in about 25 hosts (GRBs 120804A and 100206A), provides additional support to our previous conclusion that short GRBs track both stellar mass and star formation activity.
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