  (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.
