We show that Type Ia supernovae (SNe Ia) are formed within both very young and old stellar populations, with observed rates that depend on the stellar mass and mean star formation rates (SFRs) of their host galaxies. Models in which the SN Ia rate depends solely on host galaxy stellar mass are ruled out with >99% confidence. Our analysis is based on 100 spectroscopically confirmed SNe Ia, plus 24 photometrically classified events, all from the Supernova Legacy Survey (SNLS) and distributed over 0.2 < z < 0.75. We estimate stellar masses and SFRs for the SN Ia host galaxies by fitting their broadband spectral energy distributions with the galaxy spectral synthesis code PÉGASE.2. We show that the SN Ia rate per unit mass is proportional to the specific SFR of the parent galaxies - more vigorously star-forming galaxies host more SNe Ia per unit stellar mass, broadly equivalent to the trend of increasing SN Ia rate in later type galaxies seen in the local universe. Following earlier suggestions for a simple "two-component" model approximating the SN Ia rate, we find bivariate linear dependencies of the SN Ia rate on both the stellar masses and the mean SFRs of the host systems. We find that the SN Ia rate can be well represented as the sum of 5.3 ± 1.1 × 10-14 SNe yr -1M⊙-1 and 3.9 ± 0.7 × 10-4 SNe yr-1 (M⊙ yr-1) -1 of star formation. We also demonstrate a dependence of distant SN Ia light-curve shapes on star formation in the host galaxy, similar to trends observed locally. Passive galaxies, with no star formation, preferentially host faster declining/dimmer SNe Ia, while brighter events are found in systems with ongoing star formation. © 2006. The American Astronomical Society. All rights reserved.