Using HST ACS, we have obtained deep optical images reaching stars well below the oldest main-sequence turnoff in the spheroid, tidal stream, and outer disk of Andromeda. We have reconstructed the star formation history in these fields by comparing their color-magnitude diagrams to a grid of isochrones calibrated to Galactic globular clusters observed in the same bands. Each field exhibits an extended star formation history, with many stars younger than 10 Gyr but few younger than 4 Gyr. Considered together, the star counts, kinematics, and population characteristics of the spheroid argue against some explanations for its intermediate-age, metal-rich population, such as a significant contribution from stars residing in the disk or a chance intersection with the stream's orbit. Instead, it is likely that this population is intrinsic to the inner spheroid, whose highly disturbed structure is clearly distinct from the pressure-supported metal-poor halo that dominates farther from the galaxy's center. The stream and spheroid populations are similar, but not identical, with the stream's mean age being ~1 Gyr younger; this similarity suggests that the inner spheroid is largely polluted by material stripped from either the stream's progenitor or similar objects. The disk population is considerably younger and more metal-rich than the stream and spheroid populations, but not as young as the thin-disk population of the solar neighborhood; instead, the outer disk of Andromeda is dominated by stars of age 4-8 Gyr, resembling the Milky Way's thick disk. The disk data are inconsistent with a population dominated by ages older than 10 Gyr and in fact do not require any stars older than 10 Gyr.
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