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Magnesium diboride (MgB2) thin films on r-cut sapphire (r-Al2O3) single crystals were fabricated by a precursor, which was obtained at room temperature via a pulsed laser deposition (PLD) method using a Nd:YAG laser, and an in situ postannealing process. The onset superconducting transition, Tconset, and zero-resistivity transition, Tczero, were observed at 33.6 and 31.7 K, respectively, in the MgB2 thin films prepared by a Mg-rich target with a ratio of Mg:B = 3:2. The critical current density, Jc, calculated from magnetization measurements reached up to 0.9 × 106 A cm−2 at 20 K and 0 T. The broad angular Jc peak was found at 28 K when the magnetic fields were applied in a direction parallel to the film surface (θ = 90°). This could be indicative of the granular structure with randomly oriented grains. Our results demonstrate that this process is a promising candidate for the fabrication of MgB2 superconducting devices.