We report on simultaneous observations of the local starburst system Arp 299 with NuSTAR and Chandra, which provides the first resolved images of this galaxy up to energies of ~45 keV. Fitting the 3-40 keV spectrum reveals a column density of NH ~ 4 × 1024 cm-2, characteristic of a Compton-thick active galactic nucleus (AGN), and a 10-30 keV luminosity of 1.2 × 1043 erg s-1. The hard X-rays detected by NuSTAR above 10 keV are centered on the western nucleus, Arp 299-B, which previous X-ray observations have shown to be the primary source of neutral Fe-K emission. Other X-ray sources, including Arp 299-A, the eastern nucleus also thought to harbor an AGN, as well as X-ray binaries, contribute ≲ 10% to the 10-20 keV emission from the Arp 299 system. The lack of significant emission above 10 keV other than that attributed to Arp 299-B suggests that: (1) any AGN in Arp 299-A must be heavily obscured (NH > 1024 cm-2) or have a much lower luminosity than Arp 299-B and (2) the extranuclear X-ray binaries have spectra that cut-off above ~10 keV. Such soft spectra are characteristic of ultraluminous X-ray sources observed to date by NuSTAR.