We present deep Hubble Space Telescope (HST) photometry of the ultra-faint dwarf galaxy Eridanus II (Eri II). Eri II, which has an absolute magnitude of M_V = -7.1, is located at a distance of 339 kpc, just beyond the virial radius of the Milky Way. We determine the star formation history of Eri II and measure the structure of the galaxy and its star cluster. We find that a star formation history consisting of two bursts, constrained to match the spectroscopic metallicity distribution of the galaxy, accurately describes the Eri II stellar population. The best-fit model implies a rapid truncation of star formation at early times, with >80% of the stellar mass in place before z~6. A small fraction of the stars could be as young as 8 Gyr, but this population is not statistically significant; Monte Carlo simulations recover a component younger than 9 Gyr only 15% of the time, where they represent an average of 7 +/- 4% of the population. These results are consistent with theoretical expectations for quenching by reionization. The HST depth and angular resolution enable us to show that Eri IIs cluster is offset from the center of the galaxy by a projected distance of 23 +/- 3 pc. This offset could be an indication of a small (~50-75 pc) dark matter core in Eri II. Moreover, we demonstrate that the cluster has a high ellipticity of 0.31 +0.05/-0.06 and is aligned with the orientation of Eri II within 3 +/- 6 degrees, likely due to tides. The stellar population of the cluster is indistinguishable from that of Eri II itself.
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