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Gamma-ray bursts (GRBs) serve as powerful probes of the early universe, with their luminous afterglows revealing the locations and physical properties of star-forming galaxies at the highest redshifts, and potentially locating first-generation (Population III) stars. Since GRB afterglows have intrinsically very simple spectra, they allow robust redshifts from low signal-to-noise spectroscopy, or photometry. Here we present a photometric redshift of z ∼9.4 for the Swift detected GRB090429B based on deep observations with Gemini-North, the Very Large Telescope, and the GRB Optical and Near-infrared Detector. Assuming a Small Magellanic Cloud dust law (which has been found in a majority of GRB sight lines), the 90% likelihood range for the redshift is 9.06 < z < 9.52, although there is a low-probability tail toward somewhat lower redshifts. Adopting Milky Way or Large Magellanic Cloud dust laws leads to very similar conclusions, while a Maiolino law does allow somewhat lower redshift solutions, though in all cases the most likely redshift is found to be z > 7. The non-detection of the host galaxy to deep limits (Y(AB) ∼28, which would correspond roughly to 0.001L* at z = 1) in our late-time optical and infrared observations with the Hubble Space Telescope strongly supports the extreme-redshift origin of GRB090429B, since we would expect to have detected any low-z galaxy, even if it were highly dusty. Finally, the energetics of GRB 090429B are comparable to those of other GRBs and suggest that its progenitor is not greatly different from those of lower redshift bursts.