We present the star formation history (SFH) and its variations with galactocentric distance for the Local Group dwarf galaxy of Phoenix. They have been derived from a (F555W, F814W) color-magnitude diagram obtained from WFPC2@HST data, which reaches the oldest main-sequence turnoffs. The IAC-star and IAC-pop codes and the MinnIAC suite have been used to obtain the star formation rate as a function of time and metallicity, ψ(t, z). We find that Phoenix has had ongoing but gradually decreasing star formation over nearly a Hubble time. The highest level of star formation occurred from the formation of the galaxy till 10.5 Gyr ago, when 50% of the total star formation had already taken place. From that moment, star formation continues at a significant level until 6 Gyr ago (an additional 35% of the stars are formed in this time interval), and at a very low level till the present time. The chemical enrichment law shows a trend of slowly increasing metallicity as a function of time until 6-8 Gyr ago, when metallicity starts to increase steeply to the current value. We have paid particular attention to the study of the variations of the SFH as a function of radius. Young stars are found in the inner region of the galaxy only, but intermediate-age and old stars can be found at all galactocentric distances. The distribution of mass density in alive stars and its evolution with time has been studied. This study shows that star formation started at all galactocentric distances in Phoenix at an early epoch. If stars form in situ in Phoenix, the star formation onset took place all over the galaxy (up to a distance of about 400 pc from the center), but preferentially out of center regions. After that, our results are compatible with a scenario in which the star formation region envelope slowly shrinks as time goes on, possibly as a natural result of pressure support reduction as gas supply diminishes. As a consequence, the star formation stopped first (about 7-8 Gyr ago) in outer regions and the scale length of the stellar mass density distribution decreased with time. Finally, no traces of a true, old halo are apparent in Phoenix either in its stellar age distribution or in the stellar mass density distribution, at least out to 0.5 kpc (about 2.5 scale length) from the center.