We present high-energy (3-30 keV) NuSTAR observations of the nearest quasar, the ultraluminous infrared galaxy (ULIRG) Markarian 231 (Mrk 231), supplemented with new and simultaneous low-energy (0.5-8 keV) data from Chandra. The source was detected, though at much fainter levels than previously reported, likely due to contamination in the large apertures of previous non-focusing hard X-ray telescopes. The full band (0.5-30 keV) X-ray spectrum suggests the active galactic nucleus (AGN) in Mrk 231 is absorbed by a patchy and Compton-thin ( cm-2) column. The intrinsic X-ray luminosity (L0.5 - 30 keV ~ 1.0 × 1043 erg s-1) is extremely weak relative to the bolometric luminosity where the 2-10 keV to bolometric luminosity ratio is ~0.03% compared to the typical values of 2%-15%. Additionally, Mrk 231 has a low X-ray-to-optical power law slope (αOX ~ -1.7). It is a local example of a low-ionization broad absorption line quasar that is intrinsically X-ray weak. The weak ionizing continuum may explain the lack of mid-infrared [O IV], [Ne V], and [Ne VI] fine-structure emission lines which are present in sources with otherwise similar AGN properties. We argue that the intrinsic X-ray weakness may be a result of the super-Eddington accretion occurring in the nucleus of this ULIRG, and may also be naturally related to the powerful wind event seen in Mrk 231, a merger remnant escaping from its dusty cocoon.