We observed the 2019 January total lunar eclipse with the Hubble Space Telescopes STIS spectrograph to obtain the first near-UV (1700-3200 $r{A}$) observation of Earth as a transiting exoplanet. The observatories and instruments that will be able to perform transmission spectroscopy of exo-Earths are beginning to be planned, and characterizing the transmission spectrum of Earth is vital to ensuring that key spectral features (e.g., ozone, or O$_3$) are appropriately captured in mission concept studies. O$_3$ is photochemically produced from O$_2$, a product of the dominant metabolism on Earth today, and it will be sought in future observations as critical evidence for life on exoplanets. Ground-based observations of lunar eclipses have provided the Earths transmission spectrum at optical and near-IR wavelengths, but the strongest O$_3$ signatures are in the near-UV. We describe the observations and methods used to extract a transmission spectrum from Hubble lunar eclipse spectra, and identify spectral features of O$_3$ and Rayleigh scattering in the 3000-5500 r{A} region in Earths transmission spectrum by comparing to Earth models that include refraction effects in the terrestrial atmosphere during a lunar eclipse. Our near-UV spectra are featureless, a consequence of missing the narrow time span during the eclipse when near-UV sunlight is not completely attenuated through Earths atmosphere due to extremely strong O$_3$ absorption and when sunlight is transmitted to the lunar surface at altitudes where it passes through the O$_3$ layer rather than above it.