The Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS) and Cluster Lensing And Supernova survey with Hubble (CLASH) multi-cycle treasury programs with the Hubble Space Telescope (HST) have provided new opportunities to probe the rate of core-collapse supernovae (CCSNe) at high redshift, now extending to $zapprox2.5$. Here we use a sample of approximately 44 CCSNe to determine volumetric rates, $R_{CC}$, in six redshift bins in the range $0.1<z<2.5$. Together with rates from our previous HST program, and rates from the literature, we trace a more complete history of $R_{CC}(z)$, with $R_{CC}=0.72pm0.06$ yr$^{-1}$ Mpc$^{-3}$ 10$^{-4}$ $h_{70}^{3}$ at $z<0.08$, and increasing to $3.7^{+3.1}_{-1.6}$ yr$^{-1}$ Mpc$^{-3}$ 10$^{-4}$ $h_{70}^{3}$ to $zapprox2.0$. The statistical precision in each bin is several factors better than than the systematic error, with significant contributions from host extinction, and average peak absolute magnitudes of the assumed luminosity functions for CCSN types. Assuming negligible time delays from stellar formation to explosion, we find these composite CCSN rates to be in excellent agreement with cosmic star formation rate density (SFRs) derived largely from dust-corrected rest-frame UV emission, with a scaling factor of $k=0.0091pm0.0017,M^{-1}_{odot}$, and inconsistent (to $>95%$ confidence) with SFRs from IR luminous galaxies, or with SFR models that include simple evolution in the initial mass function over time. This scaling factor is expected if the fraction of the IMF contributing to CCSN progenitors is in the 8 to 50 $M_{odot}$ range. It is not supportive, however, of an upper mass limit for progenitors at $<20,M_{odot}$.
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