  We present results from the Hubble Higher-z Supernova Search, the first
space-based open field survey for supernovae (SNe). In cooperation with the
Great Observatories Origins Deep Survey, we have used the Hubble Space
Telescope with the Advanced Camera for Surveys to cover 300 square arcmin in
the area of the Chandra Deep Field South and the Hubble Deep Field North on
five separate search epochs (separated by 45 day intervals) to a limiting
magnitude of z'=26. These deep observations have allowed us to discover 42 SNe
in the redshift range 0.2 < z < 1.6. As these data span a large range in
redshift, they are ideal for testing the validity of Type Ia supernova
progenitor models with the distribution of expected ``delay times,'' from
progenitor star formation to SN Ia explosion, and the SN rates these models
predict. Through a Bayesian maximum likelihood test, we determine which
delay-time models best reproduce the redshift distribution of SNe Ia discovered
in this survey. We find that models that require a large fraction of ``prompt''
(less than 2 Gyr) SNe Ia poorly reproduce the observed redshift distribution
and are rejected at 95% confidence. We find that Gaussian models best fit the
observed data for mean delay times in the range of 3 to 4 Gyr.
