We report Keck Observatory high-resolution echelle spectra from 640-850 nm for eight stars near the faint end of the main sequence. These spectra are the highest resolution spectra of such late-type stars, and clearly resolve the TiO, VO, and atomic lines. The sample includes the field brown-dwarf candidate, BRI 0021-0214 (M9.5+). Very unexpectedly, it shows the most rapid rotation in the entire sample, ν sin i∼40 km s-1, which is 20X faster than typical field nonemission M stars. Equally surprising is that BRI 0021 exhibits no emission or absorption at Hα. We argue that this absence is not simply due to its cool photosphere, but that stellar activity declines in a fundamental way at the end of the main sequence. As it is the first very late M dwarf observed at high spectral resolution, BRI 0021 may be signaling a qualitative change in the angular momentum loss rate among the lowest mass stars. Conventionally, its rapid rotation would have marked BRI 0021 as very young, consistent with the selection effect which arises if the latest-type dwarfs are really brown dwarfs on cooling curves. In any case, it is unprecedented to find no sign of stellar activity in such a rapidly rotating convective star. We also discuss the possible conflict between this observation and the extremely strong Hα seen in another very cool star, PC 0025+0447. Extrapolation of Script M sign-L relations for BRI 0021 yields Script M sign∼0.065Script M sign⊙, and the other sample objects have expected masses near the H-burning limit. These include two Pleiades brown-dwarf candidates, four field M6 dwarfs and one late-type T Tauri star. The two Pleiades M6 dwarfs have ν sin i of 26 and 37 km s-1, Hα in emission, and radial velocities consistent with Pleiades membership. Similarly, the late-type T Tauri star has ν sin i∼30 km s-1 and Hα emission indicative of its youth. Two of the four late-type field dMe stars also exhibit rotation above 5 km s-1, consistent with expectations. BRI 0021 has no measurable absorption due to lithium, indicating that it is likely to be more massive than 0.065Script M sign⊙.