Chandra ACIS-S3 (Advanced CCD Imaging Spectrometer) observations of the nearby S0 galaxy NGC 1332 resolve much of the X-ray emission into 73 point sources, of which 37 lie within the D₂₅ isophote. The remaining galaxy emission comprises hot, diffuse gas and unresolved sources and is discussed in two companion papers. The point-source X-ray luminosity function (XLF) shows the characteristic break seen in other early-type galaxies at ~2 x 10³⁸ ergs s^-1. After applying corrections for detection incompleteness at low luminosities due to source confusion and contamination from diffuse galactic emission, the break vanishes and the data are well described as a single power law. This result casts further doubt on there being a "universal" XLF break in early-type galaxies, marking the division between neutron star and black hole systems. The logarithmic slope of the differential XLF (dN/dL), beta = 2.7 +/- 0.5, is marginally (~2.5 sigma) steeper than has been found for analogous completeness-corrected fits of other early-type galaxies but closely matches the behavior seen at high luminosities in these systems. Two of the sources within D₂₅ are ultraluminous X-ray sources (ULXs), although neither have LX > 2 x 10³⁹ ergs s^-1. The absence of very luminous ULXs in early-type galaxies suggests a break in the XLF slope at ~1-2 x 10³⁹ ergs s^-1, although the data were not of sufficient quality to constrain such a feature in NGC 1332. The sources have a spatial distribution consistent with the optical light and display a range of characteristics that are consistent with an LMXB population. The general spectral characteristics of the individual sources, as well as the composite source spectra, are in good agreement with observations of other early-type galaxies, although a small number of highly absorbed sources are seen. Two sources have very soft spectra, two show strong variability, indicating compact binary nature, and one source shows evidence of an extended radial profile. The authors do not detect a central source in NGC 1332, but find a faint (L_X = 2 +/- 1 x 10³⁸ ergs s^-1) point source coincident with the center of the companion dwarf galaxy NGC 1331. The region of sky containing NGC 1332 was observed with the ACIS instrument aboard Chandra between 2002 September 19 10:39 and September 20 02:59 UTC for a nominal ~60 ksec exposure.. This table contains the 73 bona fide X-ray compact sources detected in this observation, excluding one source centered within 1" of the galaxy centroid that is actually the central part of the diffuse galactic emission, one source within the D₂₅ isophote of the neighboring galaxy NGC 1331, and one source with no photons within the 0.5-7.0 keV band which is likely to be spurious. The spatial extent of 72 of the 73 sources is consistent with the instrumental PSF. One source (number 14) is clearly more extended than the PSF. This table was created by the HEASARC in May 2018 based on <a href="https://cdsarc.cds.unistra.fr/ftp/cats/J/ApJ/612/848">CDS Catalog J/ApJ/612/848</a> file table1.dat, the list of detected X-ray sources in the Chandra observation of NGC 1332. This is a service provided by NASA HEASARC .