Ultra-pure NaI(Tl) crystals are the key element for a model-independent verification of the long standing DAMA result and a powerful means to search for the annual modulation signature of dark matter interactions. The SABRE collaboration has been developing cutting-edge techniques for the reduction of intrinsic backgrounds over several years. In this paper we report the first characterization of a 3.4 kg crystal, named NaI-33, performed in an underground passive shielding setup at LNGS. NaI-33 has a record low $^{39}$K contamination of 4.3$pm$0.2 ppb as determined by mass spectrometry. We measured a light yield of 11.1$pm$0.2 photoelectrons/keV and an energy resolution of 13.2% (FWHM/E) at 59.5 keV. We evaluated the activities of $^{226}$Ra and $^{228}$Th inside the crystal to be $5.9pm0.6 mu$Bq/kg and $1.6pm0.3 mu$Bq/kg, respectively, which would indicate a contamination from $^{238}$U and $^{232}$Th at part-per-trillion level. We measured an activity of 0.51$pm$0.02 mBq/kg due to $^{210}$Pb out of equilibrium and a $alpha$ quenching factor of 0.63$pm$0.01 at 5304 keV. We illustrate the analyses techniques developed to reject electronic noise in the lower part of the energy spectrum. A cut-based strategy and a multivariate approach indicated a rate, attributed to the intrinsic radioactivity of the crystal, of $sim$1 count/day/kg/keV in the [5-20] keV region.
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