We report the discovery of the carbon-rich hyper metal-poor unevolved star J0815+4729. This dwarf star was selected from SDSS/BOSS as a metal-poor candidate and follow-up spectroscopic observations at medium resolution were obtained with the Intermediate dispersion Spectrograph and Imaging System (ISIS) at William Herschel Telescope and the Optical System for Imaging and low-intermediate-Resolution Integrated Spectroscopy (OSIRIS) at Gran Telescopio de Canarias. We use the FERRE code to derive the main stellar parameters, T-eff = 6215 +/- 82 K, and log g = 4.7 +/- 0.5, an upper limit to the metallicity of [Fe/H] <= -5.8, and a carbon abundance of [C/Fe] >= + 5.0, while [alpha/Fe] = 0.4 is assumed. The metallicity upper limit is based on the Ca II K line, which at the resolving power of the OSIRIS spectrograph cannot be resolved from possible interstellar calcium. The star could be the most iron-poor unevolved star known and also be among the ones with the largest overabundances of carbon. High-resolution spectroscopy of J0815+4729 will certainly help to derive other important elemental abundances, possibly providing new fundamental constraints on the early stages of the universe, the formation of the first stars, and the properties of the first supernovae.