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We present radial velocities of 33 red giants in the Leo I dSph galaxyobtained from spectra taken with the HIRES echelle spectrograph on the Keck Itelescope. These data have a mean precision of 2.2 km/s and lead to estimatesof the central velocity dispersion and systemic velocity of Leo I of 8.8 +- 1.3km/s, and 287.0 +- 1.9 km/s, respectively. The systemic velocity confirms pastresults that Leo I has an unusually large galactocentric velocity, implying thepresence of a massive dark halo in the Milky Way or an extended dark componentpervading the Local Group. The V-band (M/L) ratio of Leo I is in the range3.5-5.6. We have produced a set of models that account for the effects ofstellar evolution on the global mass-to-light ratio of a composite population.Because Leo I contains a dominant intermediate-age population, we find that theV-band mass-to-light ratio of Leo I would be in the range 6-13 if it werecomposed exclusively of old stars such as found in globular clusters. Thissuggests that Leo I probably does contain a significant dark halo. The mass ofthis halo is approximately 2 x10^7 solar masses, similar to the dark halomasses inferred for all other Galactic dSph galaxies.  Because Leo I is isolated and has passed the Milky Way at most once in thepast, external tides could not plausibly have inflated its centraldispersion tothe observed value. We also considered whether MOdified Newtonian Dynamics(MOND) could account for the internal kinematics of Leo I and conclude thatthis alternative gravitational model can account for the Leo I kinematicsadequately without requiring a dark halo.Comment: 32 pages plus 6 figs (fig 2 unavailable) plus 7 pages of tables  Accepted for Astronomical Journal, Nov 1998 "MOND" section revised because it  was pointed out by an alert reader that we used an out-of-date value for the  MOND acceleration parameter. MOND can now plausibly account for the Leo I  velocity dispersio

The Internal Kinematics of the Leo I Dwarf Spheroidal Galaxy: Dark Matter at the Fringe of the Milky Way