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We use the Atacama Large Millimeter/submillimeter Array (ALMA) in Cycle 1 to determine spectroscopic redshifts of high-redshift dusty star-forming galaxies (DSFGs) selected by their 1.4 mm continuum emission in the South Pole Telescope (SPT) survey. We present ALMA 3 mm spectral scans between 84 and 114 GHz for 15 galaxies and targeted ALMA 1 mm observations for an additional eight sources. Our observations yield 30 new line detections from CO, [C i], [N ii], H2O and NH3. We further present Atacama Pathfinder Experiment [C ii] and CO mid-J observations for seven sources for which only a single line was detected in spectral-scan data from ALMA Cycle 0 or Cycle 1. We combine the new observations with previously published and new millimeter/submillimeter line and photometric data of the SPT-selected DSFGs to study their redshift distribution. The combined data yield 39 spectroscopic redshifts from molecular lines, a success rate of >85%. Our sample represents the largest data set of its kind today and has the highest spectroscopic completeness among all redshift surveys of high-z DSFGs. The median of the redshift distribution is z = 3.9 ±0.4, and the highest-redshift source in our sample is at z = 5.8. We discuss how the selection of our sources affects the redshift distribution, focusing on source brightness, selection wavelength, and strong gravitational lensing. We correct for the effect of gravitational lensing and find the redshift distribution for 1.4 mm selected sources with a median redshift of z = 3.1 ±0.3. Comparing to redshift distributions selected at shorter wavelengths from the literature, we show that selection wavelength affects the shape of the redshift distribution.

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