The Borexino experiment, for the study of solar neutrino physics, requires radiopurity at the level of 5 x 10(-16) g/g U-238 equivalent (or 6 X 10(-9) Bq/kg) on a detector mass of many tons of scintillator. Feasibility studies are performed in a counting test facility now operating at LNGS, which consists of 4 t of liquid scintillator viewed by 100 photomultipliers and shielded by 100 t of water. The accomplishment of this goal requires the shielding liquid, water, to be at the 10(-13) g/g contamination level (1.2 X 10(-6) Bq/kg) or better. This paper describes the water purification system; it consists of a combination of several purification processes to remove particulate, radioactive ions, dissolved gases and other impurities. Residual contaminations are measured by analytical or direct-counting techniques. For radon measurement, particularly challenging at this low activity levels, a low background counting method has been developed.