We present an Application Specific Integrated Circuit (ASIC), named VEGA-1, designed and manufactured for low-power analog pulse processing of signals from Silicon Drift Detectors (SDDs). The VEGA-1 ASIC consists of an analog and a digital/mixed-signal section to achieve all the functionalities and specifications required for high-resolution X-ray spectroscopy in the energy range from 500 eV to 60 keV with low power consumption. The VEGA-1 ASIC has been designed and manufactured in 0.35-?m CMOS mixed-signal technology in single and 32-channel version with dimensions of 200??m × 500??m per channel. A minimum intrinsic ENC of 12 electrons r.m.s. at 3.6 ?s shaping time and room temperature is measured for the ASIC without detector. The VEGA-1 has been tested with Q10-SDD designed in Trieste and fabricated at FBK, with an active area of 10 mm2 and a thickness of 450 ?m. The aforementioned detector has an anode current of about 180 pA at +22?C. A minimum Equivalent Noise Charge (ENC) of 16 electrons r.m.s. at 3.0 ?s shaping time and ?30?C has been demonstrated with a total measured power consumption of 482 ?W.