We report the in-house fabrication of a high-resolution Fourier-transform spectrometer (FTS) for the spectroscopy of molecules in the gas phase at resolutions down to 0.002 cm-1 working in the spectral range from 5880 cm-1 (1.7 μm) to 15 380 cm-1 (650 nm). The FTS employs a supercontinuum as a broadband light source and a He:Ne laser with a homemade frequency-stabilization scheme as the spatial reference for the sampling of the interferogram on a constant optical path difference (OPD) grid. The sampling of the two lasers is performed at constant time intervals, and the resampling process is performed at the software level. The resampling of the interferogram on a constant OPD grid relies on cubic approximations of the He:Ne interference pattern to determine its zero-crossings. The use of an invariant in the sampling process allows us to perform on-the-fly data treatment. Both the hardware aspect and the data processing are described with, in each case, an original approach. We also report the successful coupling of the FTS with a high finesse optical cavity with effective mirror reflectivities of 99.76%, allowing us to reach sensitivities down to 6.5 × 10-8 cm-1 with a root-mean-square accuracy of 0.0017 cm-1 on the position of the Doppler-broadened transitions with a mean transition width of 0.046 cm-1 for spectra recorded at a spectral resolution of 0.015 cm-1. The sensitivity of the instrument per spectral element, once normalized, represents the best sensitivity reported in the literature for Fourier-transform incoherent broadband cavity-enhanced absorption spectroscopy with a supercontinuum light source.