We present deep HST F555W (V) and F814W (I) observations of a central field in the Local Group dwarf spheroidal (dSph) galaxy Leo I. The resulting color-magnitude diagram (CMD) reaches I~=26 and reveals the oldest ~=10-15 Gyr old turnoffs. Nevertheless, a horizontal branch is not obvious in the CMD. Given the low metallicity of the galaxy, this likely indicates that the first substantial star formation in the galaxy may have been somehow delayed in Leo I in comparison with the other dSph satellites of the Milky Way. The subgiant region is well and uniformly populated from the oldest turnoffs up to the 1 Gyr old turnoff, indicating that star formation has proceeded in a continuous way, with possible variations in intensity but no big gaps between successive bursts, over the galaxy's lifetime. The structure of the red clump of core He-burning stars is consistent with the large amount of intermediate-age population inferred from the main sequence and the subgiant region. In spite of the lack of gas in Leo I, the CMD clearly shows star formation continuing until 1 Gyr ago and possibly until a few hundred Myr ago in the central part of the galaxy.