Context: The CoRoT space mission routinely provides high-precision photometric measurements of thousands of stars that have been continuously observed for months. Aims: The discovery and characterization of the first very massive transiting planetary companion with a short orbital period is reported. Methods: A series of 34 transits was detected in the CoRoT light curve of an F3V star, observed from May to October 2007 for 152 days. The radius was accurately determined and the mass derived for this new transiting, thanks to the combined analysis of the light curve and complementary ground-based observations: high-precision radial-velocity measurements, on-off photometry, and high signal-to-noise spectroscopic observations. Results: CoRoT-Exo-3b has a radius of 1.01 ± 0.07 R_Jup and transits around its F3-type primary every 4.26 days in a synchronous orbit. Its mass of 21.66 ± 1.0 M_Jup, density of 26.4 ± 5.6 g cm-3, and surface gravity of logg = 4.72 clearly distinguish it from the regular close-in planet population, making it the most intriguing transiting substellar object discovered so far. Conclusions: With the current data, the nature of CoRoT-Exo-3b is ambiguous, as it could either be a low-mass brown-dwarf or a member of a new class of “superplanets”. Its discovery may help constrain the evolution of close-in planets and brown-dwarfs better. Finally, CoRoT-Exo-3b confirms the trend that massive transiting giant planets (M ≥ 4 M_Jup) are found preferentially around more massive stars than the Sun. The CoRoT space mission, launched on December 27th 2006, has been developed and is operating by CNES, with the contribution of Austria, Belgium, Brasil, ESA, Germany and Spain. The first CoRoT data will be available to the public in February 2009 from the CoRoT archive: http://idoc-corot.ias.u-psud.fr/ Table of the COROT photometry is only available in electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr (130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/491/889