Thin superconducting films of magnesium diboride (MgB2) with Tc approx 24K were prepared on various oxide substrates by pulsed laser deposition (PLD) followed by an in-situ anneal. A systematic study of the influence of various in-situ annealing parameters shows an optimum temperature of about 600C in a background of 0.7 atm. of Ar/4%H2 for layers consisting of a mixture of magnesium and boron. Contrary to ex-situ approaches (e.g. reacting boron films with magnesium vapor at 900C), these films are processed below the decomposition temperature of MgB2. This may prove enabling in the formation of multilayers, junctions, and epitaxial films in future work. Issues related to the improvement of these films and to the possible in-situ growth of MgB2 at elevated temperature are discussed.