Borexino is an organic liquid scintillator underground detector for low energy solar neutrinos. The experiment has to satisfy extremely stringent low background requirements. The thin nylon spherical scintillator containment vessel has to meet cleanliness and low radioactivity levels second only, within the detector, to the scintillator itself. Overall, the background from the vessel in the fiducial volume of the detector must be kept at the level of one event per day or better. The requirements, design choices, results from laboratory tests, and fabrication techniques that have been adopted to meet this goal are presented. Details of the precautions taken during the installation of the vessels inside the Borexino detector are also discussed. © 2005 American Institute of Physics