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 \times 10 pc at a distance of 1.88 kpc , masses 16,000 M _ { \sun } , and exhibits no star formation . Near-infrared background starlight polarizations were obtained for the Galactic Plane Infrared Polarization Survey using the 1.8m Perkins telescope and the Mimir instrument . The cloud area of 0.78 square degrees contains 2,684 significant starlight polarizations for 2MASS-matched stars brighter than 12.5 mag in H -band . Polarizations are generally aligned with the cloud ’ s major axis , showing an average P.A . dispersion of 15 ^ { \circ } \pm 2 ^ { \circ } and polarization of 1.8 \pm 0.6 \% . The polarizations were combined with Galactic Ring Survey ^ { 13 } CO 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 \pm 0.04 \mu G. The magnetic field strength map exhibits seven enhancements , or ‘ magnetic cores. ’ These cores show an average magnetic field strength of 8.3 \pm 0.9 \mu G , radius of 1.2 \pm 0.2 pc , intercore spacing of 5.7 \pm 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 \pm 0.02 for n _ { H _ { 2 } } \geq 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 .