The IR-active vibrational modes of single-walled carbon nanotubes have been observed by optical transmission through thin films of bundled nanotubes. Because IR-active chemical functional groups, e.g., -COOH, -OH, might be attached to the tube walls and contribute additional spectral features, we have also studied the effects of chemical purification and long-term high-temperature vacuum annealing on the IR spectrum. Through comparison with theory, we are able to assign much of the sharp structure observed in our IR spectra.