We report the discovery of the host galaxy of dark burst GRB080607 at z_GRB=3.036. GRB080607 is a unique case of a highly extinguished (A_V~3 mag) afterglow that was yet sufficiently bright for high-quality absorption-line spectroscopy. The host galaxy is clearly resolved in deep HST WF3/IR F160W images and well detected in the Spitzer IRAC 3.5 micron and 4.5 micron channels, while displaying little/no fluxes in deep optical images from Keck and Magellan. The extremely red optical-infrared colors are consistent with the large extinction seen in the afterglow light, suggesting that the large amount of dust and gas surface mass density seen along the afterglow sightline is not merely local but likely reflects the global dust content across the entire host galaxy. Adopting the dust properties and metallicity of the host ISM derived from studies of early-time afterglow light and absorption-line spectroscopy, we perform a stellar population synthesis analysis of the observed spectral energy distribution to constrain the intrinsic luminosity and stellar population of this dark burst host. The host galaxy is best described by an exponentially declining star formation rate of e-folding time tau=2 Gyr and an age of ~2 Gyr. We also derive an extinction corrected star formation rate of SFR 125 h^{-2} M_sun/yr and a total stellar mass of M_* ~ 4x10^11 h^{-2} M_sun. Our study provides an example of massive, dusty star-forming galaxies contributing to the GRB host galaxy population, supporting the notion that long-duration GRBs trace the bulk of cosmic star formation.
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