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 the OI IR triplet data (Abia & Rebolo 1989; Tomkin et al. 1992; Cavallo, Pilachowski, & Rebolo 1997). 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 plus or minus 0.11dex with respect to the abundances determined from the triplet using the same stellar parameters and model photospheres. Our new results show a smooth extension of the Edvardsson et al.'s (1993) [O/Fe] versus metallicity curve to much lower abundances. 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. It appears that unevolved metal-poor stars are better tracers of the early chemical evolution of the Galaxy. The extrapolation of our results to very low metallicities indicates that the ratio of oxygen to iron emerging from the first Type II SNe in the early Galaxy was indeed close to unity. The higher [O/Fe] ratios we find in dwarfs has an impact on the age determination of globular clusters, and suggest that current age estimates have to be reduced by about 1-2 Gyr.