  We introduce a comprehensive analysis of multi-epoch stellar line-of-sight
velocities to determine the intrinsic velocity dispersion of the ultrafaint
satellites of the Milky Way. Our method includes a simultaneous Bayesian
analysis of both membership probabilities and the contribution of binary
orbital motion to the observed velocity dispersion within a 14-parameter
likelihood. We apply our method to the Segue 1 dwarf galaxy and conclude that
Segue 1 is a dark-matter-dominated galaxy at high probability with an intrinsic
velocity dispersion of 3.7^{+1.4}_{-1.1} km/sec. The dark matter halo required
to produce this dispersion must have an average density of 2.5^{+4.1}_{-1.9}
solar mass/pc^3 within a sphere that encloses half the galaxy's stellar
luminosity. This is the highest measured density of dark matter in the Local
Group. Our results show that a significant fraction of the stars in Segue 1 may
be binaries with the most probable mean period close to 10 years, but also
consistent with the 180 year mean period seen in the solar vicinity at about 1
sigma. Despite this binary population, the possibility that Segue 1 is a bound
star cluster with the observed velocity dispersion arising from the orbital
motion of binary stars is disfavored by the multi-epoch stellar velocity data
at greater than 99% C.L. Finally, our treatment yields a projected
(two-dimensional) half-light radius for the stellar profile of Segue 1 of
28^{+5}_{-4} pc, in excellent agreement with photometric measurements.
