We present two NuSTAR observations of the local Seyfert 2 active galactic nucleus (AGN) and an ultraluminous X-ray source (ULX) candidate in NGC 5643. Together with archival data from Chandra, XMM-Newton and Swift-BAT, we perform a high-quality broadband spectral analysis of the AGN over two decades in energy ($sim$0.5-100 keV). Previous X-ray observations suggested that the AGN is obscured by a Compton-thick (CT) column of obscuring gas along our line-of-sight. However, the lack of high-quality $gtrsim$ 10 keV observations, together with the presence of a nearby X-ray luminous source, NGC 5643 X-1, had left significant uncertainties in the characterization of the nuclear spectrum. NuSTAR now enables the AGN and NGC 5643 X-1 to be separately resolved above 10 keV for the first time and allows a direct measurement of the absorbing column density toward the nucleus. The new data show that the nucleus is indeed obscured by a CT column of $N_{rm{H}}$ $gtrsim$ 5 $times$ 10$^{24}$ cm$^{-2}$. The range of 2-10 keV absorption-corrected luminosity inferred from the best fitting models is $L_{2-10,rm{int}} =$ (0.8-1.7) $times$ 10$^{42}$ erg s$^{-1}$, consistent with that predicted from multiwavelength intrinsic luminosity indicators. We also study the NuSTAR data for NGC 5643 X-1, and show that it exhibits evidence for a spectral cut-off at energy, $E$ $sim$ 10 keV, similar to that seen in other ULXs observed by NuSTAR. Along with the evidence for significant X-ray luminosity variations in the 3-8 keV band from 2003-2014, our results further strengthen the ULX classification of NGC 5643 X-1.
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