We announce the discovery of the transiting planet CoRoT-13b. Ground-based follow-up in CFHT and IAC80 confirmed CoRoT's observations. The mass of the planet was measured with the HARPS spectrograph and the properties of the host star were obtained analyzing HIRES spectra from the Keck telescope. It is a hot Jupiter-like planet with an orbital period of 4.04 days, 1.3 Jupiter masses, 0.9 Jupiter radii, and a density of 2.34 g cm-3. It orbits a G0V star with T_eff = 5 945 K, M* = 1.09 M&sun;, R_* = 1.01 R&sun;, solar metallicity, a lithium content of + 1.45 dex, and an estimated age of between 0.12 and 3.15 Gyr. The lithium abundance of the star is consistent with its effective temperature, activity level, and age range derived from the stellar analysis. The density of the planet is extreme for its mass, implies that heavy elements are present with a mass of between about 140 and 300 {M}⊕. 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. Part of the observations were obtained at the Canada-France-Hawaii Telescope (CFHT) which is operated by the National Research Council of Canada, the Institut National des Sciences de l'Univers of the Centre National de la Recherche Scientifique of France, and the University of Hawaii. Based on observations made with HARPS spectrograph on the 3.6-m European Organisation for Astronomical Research in the Southern Hemisphere telescope at La Silla Observatory, Chile (ESO program 184.C-0639). Based on observations made with the IAC80 telescope operated on the island of Tenerife by the Instituto de Astrofísica de Canarias in the Spanish Observatorio del Teide. Part of the data presented herein were obtained at the W. M. Keck Observatory, which is operated as a scientific partnership among the California Institute of Technology, the University of California and the National Aeronautics and Space Administration. The Observatory was made possible by the generous financial support of the W. M. Keck Foundation.