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Using the Infrared Spectrograph on board the Spitzer Space Telescope, we present low-resolution (64 < λ/δλ < 124), mid-infrared (20-38 μm) spectra of 23 high-redshift ULIRGs detected in the Boötes field of the NOAO Deep Wide-Field Survey. All of the sources were selected to have (1) f_ν(24 μm)>0.5 mJy; (2) R – [24]>14 Vega mag; and (3) a prominent rest frame 1.6 μm stellar photospheric feature redshifted into Spitzer's 3-8 μm IRAC bands. Of these, 20 show emission from polycyclic aromatic hydrocarbons (PAHs), usually interpreted as signatures of star formation. The PAH features indicate redshifts in the range 1.5 < z < 3.0, with a mean of = 1.96 and a dispersion of 0.30. Based on local templates, these sources have extremely large infrared luminosities, comparable to that of submillimeter galaxies. Our results confirm previous indications that the rest-frame 1.6 μm stellar bump can be efficiently used to select highly obscured star-forming galaxies at z ≈ 2, and that the fraction of starburst-dominated ULIRGs increases to faint 24 μm flux densities. Using local templates, we find that the observed narrow redshift distribution is due to the fact that the 24 μm detectability of PAH-rich sources peaks sharply at z = 1.9. We can analogously use observed spectral energy distributions to explain the broader redshift distribution of Spitzer-detected ULIRGs that are dominated by an active galactic nucleus (AGN). Finally, we conclude that z ≈ 2 sources with a detectable 1.6 μm stellar opacity feature lack sufficient AGN emission to veil the 7.7 μm PAH band.