  (ABRIDGED) 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 where 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.
Using multi-band photometry, we estimate stellar masses and SFRs for the SN Ia
host galaxies by fitting their broad-band spectral energy distributions with
the galaxy spectral synthesis code, PEGASE.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 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 slower-declining/brighter events are only
found in systems with ongoing star-formation. We model the light-curve width
distribution in star-forming galaxies as the sum of a young component, and an
old component taken from the distribution in non-star-forming galaxies.
