  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 $z\approx2.5$. 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
$R_{CC}=0.72\pm0.06$ 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 $z\approx2.0$. 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
$k=0.0091\pm0.0017\,M^{-1}_{\odot}$, 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 $<20\,M_{\odot}$.
