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 the Hubble Space Telescope/Advanced Camera for Surveys (object 3 in the work of Stanway, Bunker & McMahon 2003). The large colour of (i′−z′)AB = 1.6 indicated a spectral break consistent with the Lyman α forest absorption shortward of Lyman α at z≈ 6. The galaxy is very compact (marginally resolved with ACS with a half-light radius of 0.08 arcsec, so rhl < 0.5 h−170 kpc). We have obtained a deep (5.5 h) spectrum of this z′AB = 24.7 galaxy with the DEIMOS optical spectrograph on the Keck Telescope, and here we report the discovery of a single emission line centred on 8245 Å detected at 20σ with a flux of ƒ≈ 2 × 10−17 erg cm−2 s−1. The line is clearly resolved with detectable structure at our resolution of better than 55 km s−1, and the only plausible interpretation consistent with the ACS photometry is that we are seeing Lyman α emission from a z = 5.78 galaxy. This is the highest redshift galaxy to be discovered and studied using HST data. The velocity width (ΔvFWHM = 260 km s−1) and rest-frame equivalent width (WLyαrest = 20 Å) 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 N vλ1240- Å 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 34 h−270 M⊙ yr−1 (ΩM = 0.3, ΩΛ = 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 et al. to select star-forming galaxies atz≈ 6.