(abridged) We present kinematic results for a sample of 387 stars located near Leo I based on spectra obtained with the MMT's Hectochelle spectrograph near the MgI/Mgb lines. We estimate the mean velocity error of our sample to be 2.4 km/s, with a systematic error of < 1 km/s. We produce a final sample of 328 Leo I red giant members, from which we measure a mean heliocentric radial velocity of 282.9 +/- 0.5 km/s, and a mean radial velocity dispersion of 9.2 +/- 0.4 km/s for Leo I. The dispersion profile of Leo I is flat out to beyond its classical `tidal' radius. We 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. The mass and V-band mass-to-light ratio of Leo I estimated from equilibrium models are in the ranges 5-7 x 10^7 M_sun and 9-14 (solar units), respectively, out to 1 kpc from the galaxy center. Leo I members located outside a `break radius' (about 400 arcsec = 500 pc) exhibit significant velocity anisotropy, whereas stars interior appear to have isotropic kinematics. We propose 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. The lack of extended tidal arms in Leo I suggests the galaxy has experienced only one perigalactic passage with the Milky Way, implying that Leo I may have been injected into its present orbit by a third body a few Gyr before perigalacticon.

ApJ accepted, 23 figures, access paper as a pdf file at http://www.astro.lsa.umich.edu/~mmateo/research.html