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In this paper, an X- and gamma-ray detector based on a silicon drift detector (SDD) coupled to a CsI(Tl) scintillating crystal is presented. The SDD is operated both as a direct X-ray detector for photons interacting in silicon and as an indirect detector for photons interacting in the scintillator. As interactions in silicon and in CsI yield different amounts of charge per unit energy deposited, discrimination of the place of interaction is necessary to obtain the correct energy. Discrimination of the interaction type is carried out by means of pulse shape discrimination performed with two parallel processing chains with different shaping times. The performance of the detectors are described and discussed. Pulse shape discrimination is possible throughout the detector energy range. Index Terms--Gamma-ray detectors, pulse shape discrimination, silicon drift detectors, X-ray detectors.