  We present the deepest near-UV image of M32 to date, which for the first time
resolves hot HB stars in an elliptical galaxy. Given the near-solar metallicity
of M32, much larger than that of globular clusters, the existence of an
extended HB is a striking example of the 2nd parameter effect, and, most
importantly, provides direct evidence that hot HB stars and their progeny are
the major contributors to the UV upturn phenomenon observed in ellipticals. Our
image, obtained with STIS, detects ~8000 stars in a 25"x25" field, centered
7.7" from the galaxy nucleus. These stars span a range of 21-28 mag in the
STMAG system, and in the deepest parts of the image, our catalog is reasonably
complete (25%) to a magnitude of 27. The hot HB spans a range of 25-27 mag at
effective temperatures hotter than 8500 K. We interpret this near-UV luminosity
function with an extensive set of HB and post-HB evolutionary tracks. Although
the UV-to-optical flux ratio in M32 is weak enough to be explained solely by
the presence of post-AGB stars, our image conclusively demonstrates that it
arises from a small fraction (~5%) of the population passing through the hot HB
phase. The majority of the population presumably evolves through the red HB and
subsequent post-AGB phases; however, we see far fewer UV-bright stars than
expected from the lifetimes of canonical hydrogen-burning low-mass post-AGB
tracks. There are several possible explanations: (1) the transition from AGB to
Teff > 60000K could be much more rapid than previously thought; (2) the vast
majority of the post-AGB stars could be evolving along helium-burning tracks;
(3) the post-AGB stars could be surrounded by circumstellar dust during the
transition from the AGB to Teff > 60000K.
