  To test the idea that ultraluminous X-ray sources (ULXs) in external galaxies
represent a class of accreting intermediate-mass black holes (IMBHs), we have
undertaken a program to identify ULXs and a lower luminosity X-ray comparison
sample with the highest quality data in the {\it Chandra} archive. We establish
as a general property of ULXs that the most X-ray-luminous objects possess the
flattest X-ray spectra (in the {\it Chandra} bandpass). No prior sample studies
have established the general hardening of ULX spectra with luminosity. This
hardening occurs at the highest luminosities (absorbed luminosity
$\geq5\times10^{39}$~erg~s$^{-1}$) and is in line with recent models arguing
that ULXs are actually stellar-mass black holes. From spectral modeling, we
show that the evidence originally taken to mean that ULXs are IMBHs - i.e., the
"simple IMBH model" - is nowhere near as compelling when a large sample of ULXs
is looked at properly. During the last couple of years, {\it XMM-Newton}
spectroscopy of ULXs has to a large extent begun to negate the simple IMBH
model based on fewer objects. We confirm and expand these results, which
validates the {\it XMM-Newton} work in a broader sense with independent X-ray
data. We find that (1) cool-disk components are present with roughly equal
probability and total flux fraction for any given ULX, regardless of
luminosity, and (2) cool-disk components extend below the standard ULX
luminosity cutoff of 10$^{39}$~erg~s$^{-1}$, down to our sample limit of
10$^{38.3}$~erg~s$^{-1}$. The fact that cool disk components are not correlated
with luminosity damages the argument that cool disks indicate IMBHs in ULXs,
for which strong statistical support was never found.
