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Sapphire has mechanical and electrical properties that are advantageous for the construction of internal components of radiation detectors such as time projection chambers and bolometers. However, it has proved difficult to assess its $\rm ^{232}Th$ and $\rm ^{238}U$ content down to the picogram per gram level. This work reports an experimental verification of a computational study that demonstrates $γγ$ coincidence counting, coupled with neutron activation analysis (NAA), can reach ppt sensitivities. Combining results from $γγ$ coincidence counting with those of earlier single-$γ$ counting based NAA shows that a sample of Saint Gobain sapphire has $\rm ^{232}Th$ and $\rm ^{238}U$ concentrations of $<0.26$ ppt and $<2.3$ ppt, respectively; the best constraints on the radiopurity of sapphire.