We present results of deep echelle spectrophotometry of eight Galactic H II regions located at Galactocentric distances between 6.3 and 10.4 kpc. The data have been taken with the Very Large Telescope Ultraviolet Echelle Spectrograph in the 3100-10360 Å range. We have derived C++ and O++ abundances from recombination lines for all the objects as well as O+ abundances from this kind of line for three of the nebulae. The intensity of recombination lines is almost independent of the assumed electron temperature as well as of the possible presence of spatial temperature variations or fluctuations inside the nebulae. These data allow the determination of the gas-phase C and O abundance gradients of the Galactic disk, of paramount importance for chemical evolution models. This is the first time the C gradient is derived from such a large number of H II regions distributed in such a wide range of Galactocentric distances. Abundance gradients are found of the form Δlog(O/H)=-0.044+/-0.010 dex kpc-1, Δlog(C/H)=-0.103+/-0.018 dex kpc-1, and Δlog(C/O)=-0.058+/-0.018 dex kpc-1. Based on observations collected at the European Southern Observatory, Chile; proposals ESO 68.C-0149(A) and ESO 70.C-0008(A).