We have performed a detailed oxygen abundance analysis of 23 metal-poor (-3.0 < [Fe/H] < -0.3) unevolved halo stars and one giant through the OH bands in the near UV, using high-resolution echelle spectra. Oxygen is found to be overabundant with respect to iron in these stars, with the [O/Fe] ratio increasing from 0.6 to 1 between [Fe/H] = -1.5 and -3.0. The behavior of the oxygen overabundance with respect to [Fe/H] is similar to that seen in previous works based on O I IR triplet data. Contrary to the previously accepted picture, our oxygen abundances, derived from low-excitation OH lines, agree well with those derived from high-excitation lines of the triplet. For nine stars in common with Tomkin et al. we obtain a mean difference of 0.00 +/- 0.11 dex with respect to the abundances determined from the triplet using the same stellar parameters and model photospheres. For four stars in our sample we have found measurements of the [O I] lambda6300 line in the literature, from which we derive oxygen abundances consistent (average difference 0.09 dex) with those based on OH lines, showing that the long-standing controversy between oxygen abundances from forbidden and permitted lines in metal-poor unevolved stars can be resolved. Our new oxygen abundances show a smooth extension of the Edvardsson and coworkers [O/Fe] versus metallicity curve to much lower abundances, with a slope of -0.31 +/- 0.11 (taking into account the error bars in both oxygen abundances and metallicities) in the range -3 < [Fe/H] < -1. The extrapolation of our results to very low metallicities indicates that the first Type II supernovae in the early Galaxy provided oxygen to iron ratios of [O/Fe] >~ 1. The oxygen abundances of unevolved stars, when compared with values in the literature for giants of similar metallicity, imply that the latter may have suffered a process of oxygen depletion. As a result, unevolved metal-poor stars shall be considered better tracers of the early evolution of oxygen in the Galaxy. The higher [O/Fe] ratios we find in dwarfs has an impact on the age determination of globular clusters, suggesting that current age estimates have to be reduced by about 1-2 Gyr.