Supercontinua generated in highly nonlinear fibers by ultrashort-pulse lasers can be used for high-resolution Fourier transform absorption spectroscopy. The practical advantages of these bright ultrabroadband light sources for spectroscopy in the near-infrared region are reported. A ${\mathrm{Cr}}^{4+}:\mathrm{YAG}$ femtosecond laser broadened by an extruded soft-glass photonic crystal fiber, emitting from $1200\text{to}2200\mathrm{nm}$ and from $675\text{to}950\mathrm{nm}$ , provides a spectral radiance $1\times {10}^5$ times higher than that of a $3000\mathrm{K}$ blackbody and ${10}^2$ times higher than that of synchrotron radiation. The ${\mathrm{C}}_2{\mathrm{H}}_2$ and $\mathrm{N}{\mathrm{H}}_3$ overtone spectra are recorded by using this source within a few seconds.