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We have built a high-resolution soft X-ray spectrometer using a 3 × 3 array of superconducting Nb-Al-AlOx-Al-Nb tunnel junctions. The detector is cooled to ∼0.1 K by a two-stage adiabatic demagnetization refrigerator while being held at the end of a 40-cm-long detector snout that can be inserted into an experimental chamber. The 0.6 mm × 0.6 mm array has an energy resolution of 10-15 eV FWHM for X-ray energies below 1 keV and total count rate capabilities above 100 000 counts/s. We demonstrate the capabilities of the instrument in a study of anisotropy in c-axis oriented MgB2 films by near edge X-ray absorption spectroscopy. The absorption fine structure at the B K-edge differs for excitation with X-rays polarized along the c-axis and in the ab-plane of the MgB2 film. We discuss the implications of the anisotropy for the density of states in MgB2 at the Fermi surface.