We present deep echelle spectrophotometry of the Galactic H II region NGC 2579. The data have been taken with the Very Large Telescope Ultraviolet-Visual Echelle Spectrograph in the 3550-10 400 Å range. This object, which has been largely neglected, shows however a rather high surface brightness, a high ionization degree and is located at a galactocentric distance of 12.4 ± 0.7 kpc. Therefore, NGC 2579 is an excellent probe for studying the behaviour of the gas phase radial abundance gradients in the outer disc of the Milky Way. We derive the physical conditions of the nebula using several emission line-intensity ratios as well as the abundances of several ionic species from the intensity of collisionally excited lines. We also determine the ionic abundances of C2+, O+ and O2+ - and therefore the total O abundance - from faint pure recombination lines. The results for NGC 2579 permit to extend our previous determinations of the C, O and C/O gas phase radial gradients of the inner Galactic disc to larger galactocentric distances. We find that the chemical composition of NGC 2579 is consistent with flattened gradients at its galactocentric distance. In addition, we have built a tailored chemical evolution model that reproduces the observed radial abundance gradients of O, C and N and other observational constraints. We find that a levelling out of the star formation efficiency about and beyond the isophotal radius can explain the flattening of chemical gradients observed in the outer Galactic disc.