As part of an on-going effort to identify, understand and correct for astrophysics biases in the standardization of Type Ia supernovae (SN Ia) for cosmology, we have statistically classified a large sample of nearby SNe Ia into those that are located in predominantly younger or older environments. This classification is based on the specific star formation rate measured within a projected distance of 1 kpc from each SN location (LsSFR). This is an important refinement compared to using the local star formation rate directly, as it provides a normalization for relative numbers of available SN progenitors and is more robust against extinction by dust. We find that the SNe Ia in predominantly younger environments are ΔY = 0.163 ± 0.029 mag (5.7σ) fainter than those in predominantly older environments after conventional light-curve standardization. This is the strongest standardized SN Ia brightness systematic connected to the host-galaxy environment measured to date. The well-established step in standardized brightnesses between SNe Ia in hosts with lower or higher total stellar masses is smaller, at ΔM = 0.119 ± 0.032 mag (4.5σ), for the same set of SNe Ia. When fit simultaneously, the environment-age offset remains very significant, with ΔY = 0.129 ± 0.032 mag (4.0σ), while the global stellar mass step is reduced to ΔM = 0.064 ± 0.029 mag (2.2σ). Thus, approximately 70% of the variance from the stellar mass step is due to an underlying dependence on environment-based progenitor age. Also, we verify that using the local star formation rate alone is not as powerful as LsSFR at sorting SNe Ia into brighter and fainter subsets. Standardization that only uses the SNe Ia in younger environments reduces the total dispersion from 0.142 ± 0.008 mag to 0.120 ± 0.010 mag. We show that as environment-ages evolve with redshift, a strong bias, especially on the measurement of the derivative of the dark energy equation of state, can develop. Fortunately, data that measure and correct for this effect using our local specific star formation rate indicator, are likely to be available for many next-generation SN Ia cosmology experiments.