The ages of two old galaxies (53W091, 53W069) at high redshifts are used to constrain the value of the cosmological constant in a flat universe (Lambda CDM) and the density parameter Omega(M) in Friedmann-Robertson-Walker models with no Lambda term. In the case of Lambda CDM models, the quoted galaxies yield two lower limits for the vacuum energy density parameter, Omega(Lambda) greater than or equal to 0.42 and Omega(Lambda) greater than or equal to 0.5, respectively. Although compatible with the limits from statistics of gravitational lensing and cosmic microwave background, these lower bounds are more stringent than the ones recently determined using Type Ia supernovae as standard candles. For matter-dominated universes (Omega(Lambda) = 0), the existence of these galaxies imply that the universe is open with the matter density parameter constrained by Omega(M) less than or equal to 0.45 and Omega(M) less than or equal to 0.37, respectively. In particular, these results disagree completely with the analysis of field galaxies, which gives a lower limit Omega(M) greater than or equal to 0.40.