The Axion Resonant Interaction Detection Experiment (ARIADNE) is a nuclear-magnetic-resonance-based experiment that will search for novel axion-induced spin-dependent interactions between an unpolarized source mass rotor and spin-polarized He-3 nuclei placed nearby. To detect a feeble axion-mediated signal at the subattotesla level, the experiment relies on ultralow magnetic background and noise. We measure and characterize the magnetic field from a prototype tungsten rotor. We show that the field is dominantly caused by a few discrete magnetic dipoles, likely due to impurities in the rotor. This is done via a numerical optimization pipeline which fits for the locations and magnetic moments of each dipole. We find that under the current demagnetization procedure, the magnetic moment of the impurities is bounded at 10(-9) A m(2). We further show that a shielding factor of 10(9) will support ARIADNE's design sensitivity with the current level of tungsten purity and demagnetization process.