As part of a major program to use isolated Local Group dwarf galaxies as near-field probes of cosmology, we have obtained deep images of the dwarf irregular galaxy Leo A with the Advanced Camera for Surveys aboard the Hubble Space Telescope. From these images we have constructed a color-magnitude diagram (CMD) reaching apparent [absolute] magnitudes of (M475, M814)>~(29.0 [+4.4], 27.9 [+3.4]), the deepest ever achieved for any irregular galaxy beyond the Magellanic Clouds. We derive the star formation rate (SFR) as a function of time over the entire history of the galaxy. We find that over 90% of all the star formation that ever occurred in Leo A happened more recently than 8 Gyr ago. The CMD shows only a very small amount of star formation in the first few billion years after the big bang; a possible burst at the oldest ages cannot be claimed with high confidence. The peak SFR occurred ~1.5-4 Gyr ago, at a level 5-10 times the current value. Our modeling indicates that Leo A has experienced very little metallicity evolution; the mean inferred metallicity is consistent with measurements of the present-day gas-phase oxygen abundance. We cannot exclude a scenario in which all of the ancient star formation occurred prior to the end of the era of reionization, but it seems unlikely that the lack of star formation prior to ~8 Gyr ago was due to early loss or exhaustion of the in situ gas reservoir.