[en] BOREXINO is a large-scale liquid scintillator detector for solar neutrinos with a low expected event rate, therefore requiring an extremely low radioactive background. In this paper we present a method, which makes it possible to measure the surface and bulk activities of ²²⁶Ra in thin nylon foils for use in BOREXINO at unprecedented concentration levels, as low as 0.5 μBq/m² and 10 μBq/kg, respectively. It is based on high-sensitivity ²²²Rn emanation detection, taking advantage of the fact that the permeability of nylon for radon is strongly enhanced in the presence of water. A mathematical diffusion model is outlined to distinguish the surface and bulk activity of ²²⁶Ra from the ²²²Rn emanation rates of dry and wet film. The experimental method is described in connection with screening measurements of nylon foils of different origin. One foil could be identified with a low ²²⁶Ra contamination (corresponding to ∼10^-12 g/g ²³⁸U-equivalent) for application in BOREXINO. This film will be used for the construction of a liquid scintillator containment vessel as the central part of the BOREXINO solar neutrino detector