Context. The space telescope CoRoT searches for transiting extrasolar planets by continuously monitoring the optical flux of thousands of stars in several fields of view. Aims: We report the discovery of CoRoT-10b, a giant planet on a highly eccentric orbit (e = 0.53 ± 0.04) revolving in 13.24 days around a faint (V = 15.22) metal-rich K1V star. Methods: We used CoRoT photometry, radial velocity observations taken with the HARPS spectrograph, and UVES spectra of the parent star to derive the orbital, stellar, and planetary parameters. Results: We derive a radius of the planet of 0.97 ± 0.07 RJup and a mass of 2.75 ± 0.16 MJup. The bulk density, ρp = 3.70 ± 0.83 g cm-3, is ~2.8 that of Jupiter. The core of CoRoT-10b could contain up to 240 M_⊕ of heavy elements. Moving along its eccentric orbit, the planet experiences a 10.6-fold variation in insolation. Owing to the long circularisation time, τcirc > 7 Gyr, a resonant perturber is not required to excite and maintain the high eccentricity of CoRoT-10b. The CoRoT space mission, launched on December 27th 2006, has been developed and is operated by CNES, with the contribution of Austria, Belgium, Brazil, ESA (RSSD and Science Programme), Germany and Spain.