The observed delay-time distribution (DTD) of Type-Ia supernovae (SNe Ia) is a valuable probe of SN Ia progenitors and physics, and of the role of SNe Ia in cosmic metal enrichment. The SN Ia rate in galaxy clusters as a function of cluster redshift is an almost-direct measure of the DTD, but current estimates have been limited out to a mean redshift z=1.1, corresponding to time delays, after cluster star-formation, of over 3.2 Gyr. We analyze data from a Hubble Space Telescope monitoring project of 12 galaxy clusters at z=1.13-1.75, where we discover 29 SNe, and present their multi-band light curves. Based on the SN photometry and the apparent host galaxies, we assess cluster membership and SN type, finding 11 cases that are likely SNe Ia in cluster galaxies and 4 more cases which are possible but not certain cluster SNe Ia. We conduct simulations to estimate the SN detection efficiency, the experiments completeness, and the photometric errors, and perform photometry of the cluster galaxies to derive the cluster stellar masses. Separating the cluster sample into high-z and low-z bins, we obtain rest-frame SN Ia rates per unit formed stellar mass of $2.2 ^{+2.6}_{-1.3}times 10^{-13}{rm yr}^{-1}{rm M}_odot^{-1}$ at a mean redshift z=1.25, and $3.5^{+6.6}_{-2.8} times 10^{-13}{rm yr}^{-1}{rm M}_odot^{-1}$ at z=1.58. Combining our results with previous cluster SN Ia rates, we fit the DTD, now down to delays of 1.5 Gyr, with a power-law dependence, $t^alpha$, with $alpha=-1.30^{+0.23}_{-0.16}$. We confirm previous indications for a Hubble-time-integrated SN Ia production efficiency that is several times higher in galaxy clusters than in the field, perhaps caused by a peculiar stellar initial mass function in clusters, or by a higher incidence of binaries that will evolve into SNe Ia.
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