  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 $^{40}$K, is found to be $2.2\pm1.5$ ppb, 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 $1.20\pm0.05$
counts/day/kg/keV, which is a breakthrough since the DAMA/LIBRA experiment. Our
background model indicates that the rate is dominated by $^{210}$Pb and that
about half of this contamination is located in the PTFE 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 $\sim$0.2 counts/day/kg/keV 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.
