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We present an experimental demonstration of Additive Point Source Localization (APSL), a sparse parametric imaging algorithm that reconstructs the 3D positions and activities of multiple gamma-ray point sources. Using a handheld gamma-ray detector array and up to four $8$ $μ$Ci $^{137}$Cs gamma-ray sources, we performed both source-search and source-separation experiments in an indoor laboratory environment. In the majority of the source-search measurements, APSL reconstructed the correct number of sources with position accuracies of ${\sim}20$ cm and activity accuracies (unsigned) of ${\sim}20\%$, given measurement times of two to three minutes and distances of closest approach (to any source) of ${\sim}20$ cm. In source-separation measurements where the detector could be moved freely about the environment, APSL was able to resolve two sources separated by $75$ cm or more given only ${\sim}60$ s of measurement time. In these source-separation measurements, APSL produced larger total activity errors of ${\sim}40\%$, but obtained source separation distances accurate to within $15$ cm. We also compare our APSL results against traditional Maximum Likelihood-Expectation Maximization (ML-EM) reconstructions, and demonstrate improved image accuracy and interpretability using APSL over ML-EM. These results indicate that APSL is capable of accurately reconstructing gamma-ray source positions and activities using measurements from existing detector hardware.