  We present Magellan/IMACS spectroscopy of the recently-discovered Milky Way
satellite Eridanus II (Eri II). We identify 28 member stars in Eri II, from
which we measure a systemic radial velocity of $v_{\rm hel} = 75.6 \pm
1.3~\mbox{(stat.)} \pm 2.0~\mbox{(sys.)}~\mathrm{km\,s^{-1}}$ and a velocity
dispersion of $6.9^{+1.2}_{-0.9}~\mathrm{km\,s^{-1}}$. Assuming that Eri~II is
a dispersion-supported system in dynamical equilibrium, we derive a mass within
the half-light radius of Eri II is $1.2^{+0.4}_{-0.3} \times
10^{7}~\mathrm{M_\odot}$, indicating a mass-to-light ratio of
$420^{+210}_{-140}~\mathrm{M_\odot}/\mathrm{L_\odot}$ and confirming that it is
a dark matter-dominated dwarf galaxy. From the equivalent width measurements of
the CaT lines of 16 red giant member stars, we derive a mean metallicity of
${\rm [Fe/H]} = -2.38 \pm 0.13$ and a metallicity dispersion of $\sigma_{\rm
[Fe/H]} = 0.47 ^{+0.12}_{-0.09}$. The velocity of Eri II in the Galactic
Standard of Rest frame is $v_{\rm GSR} = -66.6~\mathrm{km\,s^{-1}}$, indicating
that either Eri II is falling into the Milky Way potential for the first time
or it has passed the apocenter of its orbit on a subsequent passage. At a
Galactocentric distance of $\sim$370 kpc, Eri II is one of the Milky Way's most
distant satellites known. Additionally, we show that the bright blue stars
previously suggested to be a young stellar population are not associated with
Eri II. The lack of gas and recent star formation in Eri II is surprising given
its mass and distance from the Milky Way, and may place constraints on models
of quenching in dwarf galaxies and on the distribution of hot gas in the Milky
Way halo. Furthermore, the large velocity dispersion of Eri II can be combined
with the existence of a central star cluster to constrain MACHO dark matter
with mass $\gtrsim10~\mathrm{M_\odot}$.
