We present the first resolved map of plane-of-sky magnetic field strength for a quiescent molecular cloud. GRSMC 45.60+0.30 subtends 40 × 10?pc at a distance of 1.88?kpc, masses 16,000 M ☉, and exhibits no star formation. Near-infrared background starlight polarizations were obtained for the Galactic Plane Infrared Polarization Survey using the 1.8?m Perkins telescope and the Mimir instrument. The cloud area of 0.78?deg2 contains 2684 significant starlight polarizations for Two Micron All Sky Survey matched stars brighter than 12.5?mag in the H band. Polarizations are generally aligned with the cloud's major axis, showing an average position angle dispersion of 15 ± 2° and polarization of 1.8 ± 0.6%. The polarizations were combined with Galactic Ring Survey 13CO spectroscopy and the Chandrasekhar-Fermi method to estimate plane-of-sky magnetic field strengths, with an angular resolution of 100 arcsec. The average plane-of-sky magnetic field strength across the cloud is 5.40 ± 0.04 μG. The magnetic field strength map exhibits seven enhancements or "magnetic cores." These cores show an average magnetic field strength of 8.3 ± 0.9 μG, radius of 1.2 ± 0.2?pc, intercore spacing of 5.7 ± 0.9?pc, and exclusively subcritical mass-to-flux ratios, implying their magnetic fields continue to suppress star formation. The magnetic field strength shows a power-law dependence on gas volume density, with slope 0.75 ± 0.02 for ≥10 cm–3. This power-law index is identical to those in studies at higher densities, but disagrees with predictions for the densities probed here.
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