We present accurate relative ages for a sample of 55 Galactic globular clusters. The ages have been obtained by measuring the difference between the horizontal branch and the turnoff in two internally photometrically homogeneous databases. The mutual consistency of the two data sets has been assessed by comparing the ages of 16 globular clusters in common between the two databases. We have also investigated the consistency of our relative age determination within the recent stellar model framework. All clusters with [Fe/H]<-1.7 are found to be old and coeval, with the possible exception of two objects, which are marginally younger. The age dispersion for the metal-poor clusters is 0.6 Gyr (rms), consistent with a null age dispersion. Intermediate-metallicity clusters (-1.7<[Fe/H]<-0.8) are on average 1.5 Gyr younger than the metal-poor ones, with an age dispersion of 1.0 Gyr (rms) and a total age range of ~3 Gyr. About 15% of the intermediate-metallicity clusters are coeval with the oldest clusters. All the clusters with [Fe/H]>-0.8 are ~1 Gyr younger than the most metal-poor ones, with a relatively small age dispersion, although the metal-rich sample is still too small to allow firmer conclusions. There is no correlation of the cluster age with the galactocentric distance. We briefly discuss the implication of these observational results for the formation history of the Galaxy. Based on observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555, and on observations made at the European Southern Observatory, La Silla, Chile, and with the Isaac Newton Group Telescopes.