We report a measurement of the Type Ia supernova (SN Ia) rate in galaxy clusters at 0.9 < z < 1.45 from the Hubble Space Telescope (HST) Cluster Supernova Survey. This is the first cluster SN Ia rate measurement with detected z > 0.9 SNe. Finding 8 +/- 1 cluster SNe Ia, we determine a SN Ia rate of 0.50 +0.23-0.19 (stat) +0.10-0.09 (sys) SNuB (SNuB = 10^-12 SNe L_{sun,B}^-1 yr^-1). In units of stellar mass, this translates to 0.36 +0.16-0.13 (stat) +0.07-0.06 (sys) SNuM (SNuM = 10^-12 SNe M_sun^-1 yr^-1). This represents a factor of approximately 5 +/- 2 increase over measurements of the cluster rate at z < 0.2. We parameterize the late-time SN Ia delay time distribution with a power law (proportional to t^s). Under the assumption of a cluster formation redshift of z_f = 3, our rate measurement in combination with lower-redshift cluster SN Ia rates constrains s = -1.41 +0.47/-0.40, consistent with measurements of the delay time distribution in the field. This measurement is generally consistent with expectations for the double degenerate scenario and inconsistent with some models for the single degenerate scenario predicting a steeper delay time distribution at large delay times. We check for environmental dependence and the influence of younger stellar populations by calculating the rate specifically in cluster red-sequence galaxies and in morphologically early-type galaxies, finding results similar to the full cluster rate. Finally, the upper limit of one host-less cluster SN Ia detected in the survey implies that the fraction of stars in the intra-cluster medium is less than 0.47 (95% confidence), consistent with measurements at lower redshifts.
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