We present the deepest near-UV image of M32 to date, which for the first time resolves hot horizontal branch (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 horizontal branch is a striking example of the second 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 elliptical galaxies. Our image, obtained with the Space Telescope Imaging Spectrograph (STIS), detects approximately 8000 stars in a 25'' × 25'' field, centered 77 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 magnitude 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-asymptotic giant branch (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 production of these hot HB stars does not appear to rely upon dynamical mechanisms—mechanisms that may play a role in the HB morphology of globular clusters. 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 > 60,000 K 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 > 60,000 K.