Supernova (SN) rates are potentially powerful diagnostics of metal enrichment and SN physics, particularly in galaxy clusters with their deep, metal-retaining potentials and relatively simple star-formation histories. We have carried out a survey for supernovae (SNe) in galaxy clusters, at a redshift range 0.5<z<0.9, using the Advanced Camera for Surveys (ACS) on the Hubble Space Telescope. We reimaged a sample of 15 clusters that were previously imaged by ACS, thus obtaining two to three epochs per cluster, in which we discovered five likely cluster SNe, six possible cluster SNe Ia, two hostless SN candidates, and several background and foreground events. Keck spectra of the host galaxies were obtained to establish cluster membership. We conducted detailed efficiency simulations, and measured the stellar luminosities of the clusters using Subaru images. We derive a cluster SN rate of 0.35 SNuB +0.17/-0.12 (statistical) ±0.13 (classification) ±0.01 (systematic) [where SNuB = SNe (100 yr 10^10 L_B_sun)^-1] and 0.112 SNuM +0.055/-0.039 (statistical) ±0.042 (classification) ±0.005 (systematic) [where SNuM = SNe (100 yr 10^10 M_sun)^-1]. As in previous measurements of cluster SN rates, the uncertainties are dominated by small-number statistics. The SN rate in this redshift bin is consistent with the SN rate in clusters at lower redshifts (to within the uncertainties), and shows that there is, at most, only a slight increase of cluster SN rate with increasing redshift. The low and fairly constant SN Ia rate out to z~1 implies that the bulk of the iron mass in clusters was already in place by z~1. The recently observed doubling of iron abundances in the intracluster medium between z=1 and 0, if real, is likely the result of redistribution of existing iron, rather than new production of iron. Comment: Accepted to ApJ. Full resolution version available at http://kicp.uchicago.edu/~kerens/HSTclusterSNe/