As part of an on-going effort to identify, understand and correct for astrophysics biases in the standardization of Type Ia supernovae (SNIa) for cosmology, we have statistically classified a large sample of nearby SNeIa into those located in predominantly younger or older environments. This classification is based on the specific star formation rate measured within a projected distance of 1kpc 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 SNeIa in predominantly younger environments are DY=0.163pm0.029 mag (5.7 sigma) fainter than those in predominantly older environments after conventional light-curve standardization. This is the strongest standardized SN Ia brightness systematic connected to host-galaxy environment measured to date. The well-established step in standardized brightnesses between SNeIa in hosts with lower or higher total stellar masses is smaller at DM=0.119pm0.032 mag (4.5 sigma), for the same set of SNeIa. When fit simultaneously, the environment age offset remains very significant, with DY=0.129pm0.032 mag (4.0 sigma), while the global stellar mass step is reduced to DM=0.064pm0.029 mag (2.2 sigma). Thus, approximately 70% of the variance from the stellar mass step is due to an underlying dependence on environment-based progenitor age. Standardization using only the SNeIa in younger environments reduces the total dispersion from 0.142pm0.008 mag to 0.120pm0.010 mag. We show that as environment ages evolve with redshift a strong bias on measurement of the dark energy equation of state parameters can develop. Fortunately, data to measure and correct for this effect is likely to be available for many next-generation experiments. [abstract shorten]