We present new kinematic results for a sample of 387 stars located in and around the Milky Way satellite dwarf spheroidal galaxy Leo I . These spectra were obtained with the Hectochelle multi-object echelle spectrograph on the MMT , and cover the MgI/Mgb lines at about 5200Å . Based on 297 repeat measurements of 108 stars , we estimate the mean velocity error ( 1 \sigma ) of our sample to be 2.4 km/s , with a systematic error of \leq 1 km/s . Combined with earlier results , we identify a final sample of 328 Leo I red giant members , from which we measure a mean heliocentric radial velocity of 282.9 \pm 0.5 km/s , and a mean radial velocity dispersion of 9.2 \pm 0.4 km/s for Leo I . The dispersion profile of Leo I is essentially flat from the center of the galaxy to beyond its classical ‘ tidal ’ radius , a result that is unaffected by contamination from field stars or binaries within our kinematic sample . We have fit the profile to a variety of equilibrium dynamical models and can strongly rule out models where mass follows light . Two-component Sersic+NFW models with tangentially anisotropic velocity distributions fit the dispersion profile well , with isotropic models ruled out at a 95 % confidence level . Within the projected radius sampled by our data ( \sim 1040 pc ) , the mass and V-band mass-to-light ratio of Leo I estimated from equilibrium models are in the ranges 5-7 \times 10 ^ { 7 } M _ { \odot } and 9-14 ( solar units ) , respectively . We find that Leo I members located outside a ‘ break radius ’ at R _ { b } \sim 400 arcsec ( 500 pc ) exhibit significant velocity anisotropy , whereas stars interior of this radius appear consistent with an isotropic velocity distribution . We propose a heuristic model in which the break radius represents the location of the tidal radius of Leo I at perigalacticon of a highly elliptical orbit . Our scenario can account for the complex star formation history of Leo I , the presence of population segregation within the galaxy , and Leo I ’ s large outward velocity from the Milky Way . Within the framework of our model , the lack of extended tidal arms in Leo I – both perpendicular to and along the line of sight – suggests the galaxy has experienced only one perigalactic passage with the Milky Way ; thus , Leo I may have been injected into its present orbit by a third body a few Gyr before perigalacticon . We discuss the plausibility of this idea within the context of hierarchical models and conclude that such an interaction is entirely possible . We also report the possible detection of a distinct kinematic structure in the Leo I field at about a 2 \sigma significance level .