We report the discovery of a luminous z=5.78 star-forming galaxy in the Chandra Deep Field South. This galaxy was selected as an `i-drop from the GOODS public survey imaging with HST/ACS (object 3 in Stanway, Bunker & McMahon 2003, astro-ph/0302212). The large colour of (i-z)_AB=1.6 indicated a spectral break consistent with the Lyman-alpha forest absorption short-ward of Lyman-alpha at z~6. The galaxy is very compact (marginally resolved with ACS with a half-light radius of 0.08arcsec, so r_hl<0.5kpc/h_70). We have obtained a deep (5.5-hour) spectrum of this z(AB)=24.7 galaxy with the DEIMOS optical spectrograph on Keck, and here we report the discovery of a single emission line centred on 8245Ang detected at 20sigma with a flux of f~2E-17 ergs/s/cm^2. The line is clearly resolved with detectable structure at our resolution of better than 55km/s, and the only plausible interpretation consistent with the ACS photometry is that we are seeing Lyman-alpha emission from a z=5.78 galaxy. This is the highest redshift galaxy to be discovered and studied using HST data. The velocity width Delta(v)_FWHM=260km/s and rest-frame equivalent width (W=20Ang) indicate that this line is most probably powered by star formation, as an AGN would typically have larger values. The starburst interpretation is supported by our non-detection of the high-ionization NV1240Ang emission line, and the absence of this source from the deep Chandra X-ray images. The star formation rate inferred from the rest-frame UV continuum is 34M_sun/yr/h^2_70 (Omega_M=0.3, Omega_Lambda=0.7). This is the most luminous starburst known at z>5. Our spectroscopic redshift for this object confirms the validity of the i-drop technique of Stanway, Bunker & McMahon (2003) to select star-forming galaxies at z~6.
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