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The CDF-S is the deepest X-ray image available and will remain so for the near future. We provide a spectroscopic (64.5%; 64% with spectral classifications) and photometric redshift catalog for the full 7 Ms sample, but much of our analysis focuses on the central (off-axis angles <5.7') region, which contains a large, faint ALMA sample of 75 >4.5-sigma 850 micron sources. We measure the 850 micron fluxes at the X-ray positions using the ALMA images, where available, or an ultradeep SCUBA-2 map. We find that the full X-ray sample produces ~10% of the 850 micron extragalactic background light. We separate the submillimeter detected X-ray sources into star-forming galaxies and AGNs using a star formation rate (SFR) versus X-ray luminosity calibration for high SFR galaxies. We confirm this separation using the X-ray photon indices. We measure the X-ray fluxes at the accurate positions of the 75 ALMA sources and detect 70% at >3-sigma in either the 0.5-2 or 2-7 keV bands. However, many of these may produce both their X-ray and submillimeter emission by star formation. Indeed, we find that only 20% of the ALMA sources have intermediate X-ray luminosities (rest-frame 8-28 keV luminosities of 10^42.5-10^44 erg/s), and none has a high X-ray luminosity (>10^44 erg/s). Conversely, after combining the CDF-S with the CDF-N, we find extreme star formation (SFR>300 solar masses per yr) in some intermediate X-ray luminosity sources but not in any high X-ray luminosity sources. We argue that the quenching of star formation in the most luminous AGNs may be a consequence of the clearing of gas in these sources.