The soft γ-ray repeater (SGR) 0526-66 is the first-identified magnetar, and is projected within the supernova remnant N49 in the Large Magellanic Cloud. Based on our ∼50 ks NuSTAR observation, we detect the quiescent-state 0526-66 for the first time in the 10-40 keV band. Based on the joint analysis of our NuSTAR and the archival Chandra ACIS data, we firmly establish the presence of the nonthermal component in the X-ray spectrum of 0526-66 in addition to the thermal emission. In the best-fit blackbody (BB) plus power-law (PL) model, the slope of the PL component (photon index Γ = 2.1) is steeper than those (Γ 1.5) for other magnetars. The soft part of the X-ray spectrum can be described with a BB component with the temperature of kT = 0.43 keV. The best-fit radius (R = 6.5 km) of the X-ray-emitting area is smaller than the canonical size of a neutron star. If we assume an underlying cool BB component with the canonical radius of R = 10 km for the neutron star in addition to the hot BB component (2BB + PL model), a lower BB temperature of kT = 0.24 keV is obtained for the passively cooling neutron star's surface, while the hot spot emission with kT = 0.46 keV dominates the thermal spectrum (∼85% of the thermal luminosity in the 0.5-5 keV band). The nonthermal component (Γ ∼ 1.8) is still required.