  The study of the dominant population of high redshift IR-luminous galaxies
(10^11 - 10^12 Lsun at 1<z<3), requires observation of sources at the ~0.1 mJy
level in the mid-IR. We present the deepest spectra taken to date with the
Infrared Spectrograph (IRS) on the Spitzer Space Telescope. We targeted two
faint (f24~0.15 mJy) sources in the Southern GOODS field at z=1.09 and z=2.69.
Spectra of the lower redshift target were taken in the observed-frame 8--21
micron range, while the spectrum of the higher redshift target covered 21--37
microns. We also present the spectra of two secondary sources within the slit.
We detect strong PAH emission in all four targets, and compare the spectra to
those of local galaxies observed by the IRS. The z=1.09 source appears to be a
typical, star-formation dominated IR-luminous galaxy, while the z=2.69 source
is a composite source with strong star formation and a prominent AGN. The IRAC
colors of this source show no evidence of rest-frame near-infrared stellar
photospheric emission. We demonstrate that an AGN which contributes only a
small (~10%) fraction of the bolometric luminosity can produce enough hot dust
emission to overwhelm the near-IR photospheric emission from stars. Such
sources would be excluded from photometric surveys which rely on the near-IR
bump to identify starbursts, leading to an underestimate of the star formation
rate density.
