  We present a quantitative analysis of the star formation history (SFH) of the
Local Group dSph galaxy Leo I, from the information in its HST [(V-I),I]
color-magnitude diagram (CMD). The method we use is based in comparing, via
synthetic CMDs, the expected distribution of stars in the CMD for different
evolutionary scenarios, with the observed distribution. We consider the SFH to
be composed by the SFR(t), the Z(t), the IMF, and a function $\beta(f,q)$,
controlling the fraction $f$ and mass ratio distribution $q$ of binary stars.
The comparison between the observed CMD and the model CMDs is done through
chi-square minimization of the differences in the number of stars in a set of
regions of the CMD.
  Our solution for the SFH of Leo I defines a minimum of chi-square in a well
defined position of the parameter space, and the derived SFR(t) is robust, in
the sense that its main characteristics are unchanged for different
combinations of the remaining parameters. However, only a narrow range of
assumptions for Z(t), IMF and $\beta(f,q)$ result in a good agreement between
the data and the models, namely: Z=0.0004, a Kroupa et al. (1993) IMF or
slightly steeper, and a relatively large fraction of binary stars. Most star
formation activity (70% to 80%) occurred between 7 and 1 Gyr ago. At 1 Gyr ago,
it abruptly dropped to a negligible value, but seems to have been active until
at least ~ 300 Myr ago. Our results don't unambiguously answer the question of
whether Leo I began forming stars around 15 Gyr ago, but it appears that the
amount of this star formation, if existing at all, would be small.
