Observations of transiting extrasolar planets are of key importance to our understanding of planets because their mass, radius, and mass density can be determined. The CoRoT space mission allows us to achieve a very high photometric accuracy. By combining CoRoT data with high-precision radial velocity measurements, we derive precise planetary radii and masses. We report the discovery of CoRoT-19b, a gas-giant planet transiting an old, inactive F9V-type star with a period of four days. After excluding alternative physical configurations mimicking a planetary transit signal, we determine the radius and mass of the planet by combining CoRoT photometry with high-resolution spectroscopy obtained with the echelle spectrographs SOPHIE, HARPS, FIES, and SANDIFORD. To improve the precision of its ephemeris and the epoch, we observed additional transits with the TRAPPIST and Euler telescopes. Using HARPS spectra obtained during the transit, we then determine the projected angle between the spin of the star and the orbit of the planet. We find that the host star of CoRoT-19b is an inactive F9V-type star close to the end of its main-sequence life. The host star has a mass M*=1.21+/-0.05 Msun and radius R*=1.65+/-0.04 Rsun. The planet has a mass of Mp=1.11+/-0.06 Mjup and radius of Rp=1.29+/-0.03 Rjup. The resulting bulk density is only rho=0.71+/-0.06 gcm-3, which is much lower than that for Jupiter. The exoplanet CoRoT-19b is an example of a giant planet of almost the same mass as Jupiter but a 30% larger radius.