We have developed a system to quantitatively measure the permeation of gases through thin flexible substrates with high sensitivity. The system consists of two chambers, a high pressure side and an ultrahigh vacuum (UHV) side, separated by the flexible sample to be analyzed. The system is calibrated using a combination of a National Institute of Standards and Technology traceable calibrated He leak and a variable aperture calibrated orifice. The base total pressure for the UHV side is 1-3x10(-10) Torr. The partial pressure of individual gases that we are studying is <10(-10) Torr. The sample to be measured is secured between the two chambers using two 2.75 conflat flanges, two copper gaskets, and two indium "O" rings. The key factors that impact sensitivity and quantification are (1) reducing the residual partial pressure of the gas of interest to as low a value as possible on the UHV side of the system, (2) sealing the plastic sample between the two chambers with no detectable gas leakage around the sample or from the outside, (3) supporting the plastic substrate so that it can withstand 700 Torr pressure on the high pressure side, and (4) developing a calibration procedure that closely mimics the actual permeability measurement. The system allows us to measure permeation rates as low as 1x10(-6) g/m(2)-day for He, 1x10(-6) g/m(2)-day for O-2, and 5x10(-7) g/m(2)-day for Ar. (C) 2005 American Institute of Physics.