Several problems in detector design, operation, and data evaluation have been encountered in the analysis by gamma-ray spectroscopy of samples containing very low levels of radioactivity. Some of these problems are: the compensation for variations in the background intensity during counting, the determination of how large the anticoincidence mantle and/or shield should be to reduce the background, the determination of the optimum counting times of the sample and background, and the compensation for spectral shifts occurring during the long counting times required for these samples. This work describes possible methods for satisfactorily solving these problems. A spectrum-resolution program, written in FORTRAN 62, 63, II, or IV, which can be used to analyze these low-activity samples for isotopic constituents, is available from the author.