Spectrometers for the lowest-level radiometric measurements require materials of extreme radiopurity. Measurements of rare nuclear decays, e.g., neutrinoless double-beta decay, can require construction and shielding materials with bulk radiopurity reaching one micro-Becquerel per kilogram or less. When such extreme material purity is achieved, surface contamination, particularly solid daughters in the natural radon decay chains, can become the limiting background. High-purity copper is an important material for ultra-low-background spectrometers and thus is the focus of this work. A method for removing surface contamination at very low levels without attacking the bulk material is described. An assay method using a low-background proportional counter made of the material under examination is employed, and the preliminary result of achievable surface contamination levels is presented.