A low-background germanium spectrometer has been installed and is being operated in an ultra-low background shield (the Gator facility) at the Gran Sasso underground laboratory in Italy (LNGS). With an integrated rate of ~ 0.16 events/min in the energy range between 100-2700 keV, the background is comparable to those of the world's most sensitive germanium detectors. After a detailed description of the facility, its background sources as well as the calibration and efficiency measurements are introduced. Two independent analysis methods are described and compared using examples from selected sample measurements. The Gator facility is used to screen materials for XENON, GERDA, and in the context of next-generation astroparticle physics facilities such as DARWIN.