  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.
