  We present the first measurement of the rate of Type Ia supernovae at high
redshift. The result is derived using a large subset of data from the Supernova
Cosmology Project. Three supernovae were discovered in a surveyed area of 1.7
square degrees. The survey spanned a $\sim 3$ week baseline and used images
with $3\sigma$ limiting magnitude of $R\sim 23$. We present our methods for
estimating the numbers of galaxies and the number of solar luminosities to
which the survey is sensitive, and the supernova detection efficiency which is
used to determine the control time, the effective time for which the survey is
sensitive to a Type Ia event. We derive a rest-frame Type Ia supernova rate at
$z\sim0.4$ of $0.82\ {^{+0.54}_{-0.37}}\ {^{+0.37}_{-0.25}} $ $h^2$ SNu (1 SNu
= 1 SN per century per $10^{10}$\Lbsun), where the first uncertainty is
statistical and the second includes systematic effects. For the purposes of
observers, we also determine the rate of SNe, per sky area surveyed, to be $
34.4\ {^{+23.9}_{-16.2}}$ SNe\ $\rm year^{-1} deg^{-2}$ for SN magnitudes in
the range $21.3 < R < 22.3$.
