We explore the idea that the Type Ia supernova (SN Ia) rate consists of two components: a prompt piece that is proportional to the star formation rate (SFR), and an extended piece that is proportional to the total stellar mass. We fit the parameters of this model to the local observations by Mannucci and collaborators and then study its impact on three important problems. On cosmic scales, the model reproduces the observed SN Ia rate density below z = 1 and predicts that it will track the measured SFR density at higher redshift, reaching a value of (1-3.5) × 10-4 yr-1 Mpc-3 at z = 2. In galaxy clusters, a large prompt contribution helps explain the Fe content of the intracluster medium. Within the Galaxy, the model reproduces the observed stellar [O/Fe] abundance ratios if we allow a short (≈0.7 Gyr) delay in the prompt component. Ongoing medium-z SN surveys will yield more accurate parameters for our model.