We present a new measurement of the volumetric rate of SNe Ia up to a redshift of 1.7, using the HST GOODS data combined with an additional HST data set covering the GOODS-North field collected in 2004. We employ a novel technique that does not require spectroscopic data for identifying SNe Ia (although spectroscopic measurements of redshifts are used for over half the sample); instead, we employ a Bayesian approach using only photometric data to calculate the probability that an object is an SN Ia. This Bayesian technique can easily be modified to incorporate improved priors on SN properties, and it is well-suited for future high-statistics SN searches in which spectroscopic follow-up of all candidates will be impractical. Here the method is validated on both ground- and space-based SN data having some spectroscopic follow-up. We combine our volumetric rate measurements with low-redshift SN data and fit to a number of possible models for the evolution of the SN Ia rate as a function of redshift. The data do not distinguish between a flat rate at redshift >0.5 and a previously proposed model, in which the Type Ia rate peaks at redshift ~1 due to a significant delay from star formation to the SN explosion. Except for the highest redshifts, where the signal-to-noise ratio is generally too low to apply this technique, this approach yields uncertainties that are smaller than or comparable to previous work. Based on observations with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by AURA, Inc., under NASA contract NAS 5-26555, under programs GO-9583, GO-9425, GO-9727, and GO-9728