Quantitative phase retrieval is experimentally demonstrated using the Inverse Compton Scattering X-ray source available at the Accelerator Test Facility (ATF) in the Brookhaven National Laboratory. Phase-contrast images are collected using in-line geometry, with a single X-ray pulse of approximate duration of one picosecond. The projected thickness of homogeneous samples of various polymers is recovered quantitatively from the time-averaged intensity of transmitted X-rays. The data are in good agreement with the expectations showing that ATF Inverse Compton Scattering source is suitable for performing phase-sensitive quantitative X-ray imaging on the picosecond scale. The method shows promise for quantitative imaging of fast dynamic phenomena.