More than 280 impulsive solar electron bursts, largely associated with type III radio bursts, have been observed by the solar wind experiment on ACE at energies below 1.4 keV. Here we illustrate some of the characteristics of these events with detailed observations of a particularly intense and long-lasting (>19 hours) event that extended smoothly as a power law in energy down to at least 142 eV. Such low-energy extensions of burst spectra may be evidence that the electrons in these bursts are accelerated at altitudes considerably greater than 1 solar radius above the photosphere. If so, then other characteristics of these events make it appear unlikely that the bursts result from shock acceleration. An alternate possibility is that acceleration actually occurs relatively low in the corona, as would be inferred from observations of type III bursts, and is followed by a more gradual release of electrons to the heliosphere.