Recent LTE analyses (Israelian et al.; Boesgaard et al.) of the OH bands in the optical ultraviolet spectra of nearby metal-poor subdwarfs indicate that oxygen abundances are generally higher than those previously determined from the analysis of the [O I] doublet in the spectra of low-metallicity giants. On average, the difference increases with decreasing metallicity and reaches Delta[O/Fe] similar to +0.6 dex as [Fe/H] approaches -3.0. Employing high-resolution (R = 50,000), high-signal-to-noise ratio (similar to 250) echelle spectra of the two stars found by Israelian et al. to have the highest [O/Fe] ratios, viz., BD +23 degrees 3130 and ED + 37 degrees 1458, we conducted abundance analyses based on about 60 Fe I and seven-nine Fe II lines. We determined from Kurucz LTE models the appropriate values of T-eff, log g, [Fe/H], and v(t), as well as abundances of Na, Ni, and the traditional alpha-elements Mg, Si, Ca, and Ti, independent of the calibration of color versus T-eff scales. We determined oxygen abundances from spectral synthesis of the stronger line (lambda 6300) of the [O I] doublet. The ionization equilibrium of Fe indicates that these two stars are subgiants rather than subdwarfs, and our derived values of T-eff are 150-300 K lower than those assumed by the previous investigators, although the resulting [Fe/H] values differ only slightly. The syntheses of the [O I] line lead to smaller values of [O/Fe], consistent with those found earlier among halo field and globular cluster giants. We obtain [O/Fe] = +0.35 +/- 0.2 for BD +23 degrees 3130 and + 0.50 +/- 0.2 for BD + 37 degrees 1458. In the former, the [O I] line is very weak (similar to 1 m Angstrom), so that the quoted [O/Fe] value may in reality be an upper limit. Therefore, in these two stars a discrepancy exists between the [O/Fe] ratios derived from [O I] and the OH feature, and the origin of this difference remains unclear. Until the matter is clarified, we suggest it is premature to conclude that the ab initio oxygen abundances of old, metal-poor stars need to be revised drastically upward.