We present a measurement of the volumetric Type Ia supernova (SN Ia) rate out to z similar or equal to 1.6 from the Hubble Space Telescope Cluster Supernova Survey. In observations spanning 189 orbits with the Advanced Camera for Surveys we discovered 29 SNe, of which approximately 20 are SNe Ia. Twelve of these SNe Ia are located in the foregrounds and backgrounds of the clusters targeted in the survey. Using these new data, we derive the volumetric SN Ia rate in four broad redshift bins, finding results consistent with previous measurements at z greater than or similar to 1 and strengthening the case for an SN Ia rate that is greater than or similar to 0.6 x 10(-4) h(70)(3) yr(-1) Mpc(-3) at z similar to 1 and flattening out at higher redshift. We provide SN candidates and efficiency calculations in a form that makes it easy to rebin and combine these results with other measurements for increased statistics. Finally, we compare the assumptions about host-galaxy dust extinction used in different high-redshift rate measurements, finding that different assumptions may induce significant systematic differences between measurements.