A cavity-enhanced absorption setup employing an incoherent broadband light source was used in combination with a Fourier-transform spectrometer to measure the spin-forbidden B-band of gaseous oxygen at approximately 688 nm and several weak absorption transitions of water vapor in the same spectral region at room temperature in ambient air. The experiments demonstrate that the sensitivity of a Fourier-transform spectrometer can be significantly improved by increasing the effective path length, while retaining a rather small sample volume. In comparison with a single-pass absorption measurement, we report a path-length enhancement factor of 200 and an improvement of the signal-to-noise ratio of approximately 6 in the present cavity-enhanced absorption experiment. The practical advantages and limitations of this novel approach are outlined and potential applications are briefly discussed.