Eridanus II (Eri II) is an ultrafaint dwarf (UFD) galaxy (MV = -7.1) located at a distance close to the Milky Way virial radius. Early shallow color-magnitude diagrams (CMDs) indicated that it possibly hosted an intermediate-age or even young stellar population, which is unusual for a galaxy of this mass. In this paper, we present new Hubble Space Telescope/Advanced Camera for Surveys CMDs reaching the oldest main-sequence turnoff with excellent photometric precision and derive a precise star formation history (SFH) for this galaxy through CMD fitting. This SFH shows that the bulk of the stellar mass in Eri II formed in an extremely short star formation burst at the earliest possible time. The derived star formation rate profile has a width at half maximum of 500 Myr and reaches a value compatible with null star formation 13 Gyr ago. However, tests with mock stellar populations and with the CMD of the globular cluster M92 indicate that the star formation period could be shorter than 100 Myr. From the quantitative determination of the amount of mass turned into stars in this early star formation burst ( ∼2 × 105 M⊙) we infer the number of supernova (SN) events and the corresponding energy injected into the interstellar medium. For reasonable estimates of the Eri II virial mass and values of the coupling efficiency of the SN energy, we conclude that Eri II could be quenched by SN feedback alone, thus casting doubts on the need to invoke cosmic reionization as the preferred explanation for the early quenching of old UFD galaxies. * Based on observations made with the NASA/ESA Hubble Space Telescope, obtained at the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5-26555. These observations are associated with programs 14224 and 14234.