The most accurate ages for the oldest stars are those obtained for nearby halo subgiants because they depend almost entirely on just the measured parallaxes and absolute oxygen abundances. In this study, we have used the Fine Guidance Sensors on the Hubble Space Telescope to determine trigonometric parallaxes, with precisions of 2.1% or better, for the Population II subgiants HD 84937, HD 132475, and HD 140283. High quality spectra have been used to derive their surface abundances of O, Fe, Mg, Si, and Ca, which are assumed to be 0.1–0.15 dex less than their initial abundances due to the effects of diffusion. Comparisons of isochrones with the three subgiants on the (log Teff, MV) diagram yielded ages of 12.08 ± 0.14, 12.56 ± 0.46, and 14.27 ± 0.38 Gyr for HD 84937, HD 132475, and HD 140283, in turn, where each error bar includes only the parallax uncertainty. The total uncertainty is estimated to be ∼ ± 0.8 Gyr (larger in the case of the near-turnoff star HD 84937). Although the age of HD 140283 is greater than the age of the universe as inferred from the cosmic microwave background by ∼0.4–0.5 Gyr, this discrepancy is at a level of <1σ. Nevertheless, the first Population II stars apparently formed very soon after the Big Bang. (Stellar models that neglect diffusive processes seem to be ruled out as they would predict that HD 140283 is ∼1.5 Gyr older than the universe.) The field halo subgiants appear to be older than globular clusters of similar metallicities: if distances close to those implied by the RR Lyrae standard candle are assumed, M 92 and M 5 are younger than HD 140283 and HD 132475 by ∼1.5 and ∼1.0 Gyr, respectively.