We are studying the Na-O anticorrelation in several globular clusters of different Horizontal Branch (HB) morphology in order to derive a possible relation between (primordial) chemical inhomogeneities and morphological parameters of the cluster population. We used the multifiber spectrograph FLAMES on the ESO Very Large Telescope UT2 and derived atmospheric parameters and elemental abundances of Fe, O and Na for about 150 red giant stars in the Galactic globular cluster NGC 6752. The average metallicity we derive is [Fe/H]=-1.56, in agreement with other results from red giants, but lower than obtained for dwarfs or early subgiants. In NGC 6752 there is not much space for an intrinsic spread in metallicity: on average, the rms scatter in [Fe/H] is 0.037+/-0.003dex, while the scatter expected on the basis of the major error sources is 0.039+/-0.003dex. The distribution of stars along the Na-O anticorrelation is different to what was found in the first paper of this series for the globular cluster NGC 2808: in NGC 6752 it is skewed toward more Na-poor stars, and it resembles more the one in M 13. Detailed modeling is required to clarify whether this difference may explain the very different distributions of stars along the HB.