We present near infrared polarimetry data of background stars shining through a selection of starless cores taken in the K band , probing visual extinctions up to A _ { V } \sim 48 .
We find that P _ { K } / { \tau _ { K } } continues to decline with increasing A _ { V } with a power law slope of roughly -0.5 .
Examination of published submillimeter ( submm ) polarimetry of starless cores suggests that by A _ { V } \gtrsim 20 the slope for P vs . \tau becomes \sim - 1 , indicating no grain alignment at greater optical depths .
Combining these two data sets , we find good evidence that , in the absence of a central illuminating source , the dust grains in dense molecular cloud cores with no internal radiation source cease to become aligned with the local magnetic field at optical depths greater than A _ { V } \sim 20 .
A simple model relating the alignment efficiency to the optical depth into the cloud reproduces the observations well .