We demonstrate broadband and sensitive cavity ring-down spectroscopy using a near infrared frequency comb and a time-resolved Fourier transform spectrometer. The cavity decays are measured simultaneously at each optical path difference and spectrally sorted, leading to purely exponential decays for each spectral element. The absorption spectra of atmospheric water and carbon dioxide are retrieved and demonstrate the high frequency resolution and absorption precision of the technique. The experimental apparatus, the measurement concept and the data treatment are described. The technique benefits from the advantages of cavity ring-down spectroscopy, i.e. the retrieved absorption does not depend on the cavity parameters, opening up for high accuracy absorption spectroscopy entirely calibration-free.