  We investigate the kinematic properties and stellar population of the
Galactic satellite Willman 1 (Wil 1) by combining Keck/DEIMOS spectroscopy with
KPNO mosaic camera imaging. Wil 1 is an ultra-low luminosity Milky Way
companion. This object lies in a region of size-luminosity space (M_V ~ -2 mag,
d ~ 38 kpc, r_half ~ 20 pc) also occupied by the Galactic satellites Bo\"otes
II and Segue 1 and 2, but no other known old stellar system. We use kinematic
and color-magnitude criteria to identify 45 stars as possible members of Wil 1.
With a systemic velocity of v_helio = -12.8 +/- 1.0 km/s, Wil 1 stars have
velocities similar to those of foreground Milky Way stars. Informed by
Monte-Carlo simulations, we identify 5 of the 45 candidate member stars as
likely foreground contaminants. We confirm a significant spread in the
abundances of the likely Wil 1 red giant branch members ([Fe/H] = -1.73 +/-
0.12 and -2.65 +/- 0.12, [Ca/Fe] = -0.4 +/- 0.18 and +0.13 +/- 0.28). This
spread supports the scenario that Wil 1 is an ultra-low luminosity dwarf galaxy
rather than a star cluster. Wil 1's innermost stars move with radial velocities
offset by 8 km/s from its outer stars and have a velocity dispersion consistent
with 0 km/s, suggesting that Wil 1 may not be in dynamical equilibrium. The
combination of the foreground contamination and unusual kinematic distribution
make it difficult to robustly determine the dark matter mass of Wil 1. As a
result, X-ray or gamma-ray observations of Wil 1 that attempt to constrain
models of particle dark matter using an equilibrium mass model are strongly
affected by the systematics in the observations presented here. We conclude
that, despite the unusual features in the Wil 1 kinematic distribution,
evidence indicates that this object is, or at least once was, a dwarf galaxy.
