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We present rest-frame far-infrared (FIR) and optical observations of the host galaxy of GRB 090423 at z = 8.23 from the Atacama Large Millimeter Array (ALMA) and the Spitzer Space Telescope, respectively. The host remains undetected to 3σ limits of Fν(222 GHz) ≲ 33 μJy and Fν(3.6 μm) ≲ 81 nJy. The FIR limit is about 20 times fainter than the luminosity of the local ULIRG Arp 220 and comparable to the local starburst M 82. Comparing this with model spectral energy distributions, we place a limit on the infrared (IR) luminosity of LIR(8-1000 μm) ≲ 3 × 1010L☼, corresponding to a limit on the obscured star formation rate of M☼ yr–1. For comparison, the limit on the unobscured star formation rate from Hubble Space Telescope rest-frame ultraviolet (UV) observations is SFRUV ≲ 1 M☼ yr–1. We also place a limit on the host galaxy stellar mass of M* ≲ 5 × 107M☼ (for a stellar population age of 100 Myr and constant star formation rate). Finally, we compare our millimeter observations to those of field galaxies at z ≳ 4 (Lyman break galaxies, Lyα emitters, and submillimeter galaxies) and find that our limit on the FIR luminosity is the most constraining to date, although the field galaxies have much larger rest-frame UV/optical luminosities than the host of GRB 090423 by virtue of their selection techniques. We conclude that GRB host galaxies at z ≳ 4, especially those with measured interstellar medium metallicities from afterglow spectroscopy, are an attractive sample for future ALMA studies of high redshift obscured star formation. [ABSTRACT FROM AUTHOR]