We develop, implement, and characterize an enhanced data reduction approach which delivers precise, accurate, radial velocities from moderate resolution spectroscopy with the fiber-fed VLT/FLAMES+GIRAFFE facility. This facility, with appropriate care, delivers radial velocities adequate to resolve the intrinsic velocity dispersions of the very faint dwarf spheroidal (dSph) galaxies. Importantly, repeated measurements let us reliably calibrate our individual velocity errors (0.2km/s<={delta}_V_<=5km/s) and directly detect stars with variable radial velocities. We show, by application to the Bootes I dSph, that the intrinsic velocity dispersion of this system is significantly below 6.5km/s reported by previous studies. Our data favor a two-population model of Bootes I, consisting of a majority "cold" stellar component, with velocity dispersion 2.4^+0.9^_-0.5_km/s, and a minority "hot" stellar component, with velocity dispersion ~9km/s, although we cannot completely rule out a single component distribution with velocity dispersion 4.6^0.8^_-0.6_km/s. We speculate that this complex velocity distribution actually reflects the distribution of velocity anisotropy in Bootes I, which is a measure of its formation processes.