Using the High Resolution Camera on board the Chandra X-ray Observatory, we have measured an accurate position for the bright persistent neutron star X-ray binary and atoll source GX 3+1. At a location that is consistent with this new position, we have discovered the near-infrared (NIR) counterpart to GX 3+1 in images taken with the PANIC and FourStar cameras on the Magellan Baade Telescope. The identification of this Ks = 15.8 ± 0.1 mag star as the counterpart is based on the presence of a Br γ emission line in an NIR spectrum taken with the Folded-port InfraRed Echelette spectrograph on the Baade Telescope. The absolute magnitude derived from the best available distance estimate to GX 3+1 indicates that the mass donor in the system is not a late-type giant. We find that the NIR light in GX 3+1 is likely dominated by the contribution from a heated outer accretion disk. This is similar to what has been found for the NIR flux from the brighter class of Z sources, but unlike the behavior of atolls fainter (LX ≈ 1036-1037 erg s-1) than GX 3+1, where optically thin synchrotron emission from a jet probably dominates the NIR flux. This paper includes data gathered with the 6.5 m Magellan Telescopes located at Las Campanas Observatory, Chile.