We present new results on the radiopurity of a 3.4-kg NaI(Tl) crystal scintillator operated in the SABRE proof-of-principle detector setup. The amount of potassium contamination, determined by the direct counting of radioactive , is found to be , lowest ever achieved for NaI(Tl) crystals. With the active veto, the average background rate in the crystal in the 1–6 keV energy region of interest (ROI) is , which is a breakthrough since the DAMA/LIBRA experiment. Our background model indicates that the rate is dominated by and that about half of this contamination is located in the polytetrafluoroethylene reflector. We discuss ongoing developments of the crystal manufacture aimed at the further reduction of the background, including data from purification by zone refining. A projected background rate lower than in the ROI is within reach. These results represent a benchmark for the development of next-generation NaI(Tl) detector arrays for the direct detection of dark matter particles.