The unprecedented sensitivity of the Infrared Spectrograph on the Spitzer Space Telescope allows for the first time the measurement of mid-infrared spectra from 14 to 38 microns of faint high-z galaxies. This unique capability is demonstrated with observations of sources having 16 micron fluxes of 3.6 mJy (CFRS 14.1157) and 0.35 mJy (CFRS 14.9025). A spectral-fitting technique is illustrated which determines the redshift by fitting emission and absorption features characteristic of nearby galaxies to the spectrum of an unknown source. For CFRS 14.1157, the measured redshift is z = 1.00+/-0.20 in agreement with the published result of z = 1.15. The spectrum is dominated by emission from an AGN, similar to the nucleus of NGC 1068, rather than a typical starburst with strong PAH emission like M82. Such spectra will be crucial in characterizing the nature of newly discovered distant galaxies, which are too faint for optical follow-up.