  We present 870 $\mu$m ALMA polarization observations of thermal dust emission
from the iconic, edge-on debris disk $\beta$ Pic. While the spatially resolved
map does not exhibit detectable polarized dust emission, we detect polarization
at the $\sim$3$\sigma$ level when averaging the emission across the entire
disk. The corresponding polarization fraction is $P_\textrm{frac}$ = $0.51 \pm
0.19$%. The polarization position angle $\chi$ is aligned with the minor axis
of the disk, as expected from models of dust grains aligned via radiative
alignment torques (RAT) with respect to a toroidal magnetic field ($B$-RAT) or
with respect to the anisotropy in the radiation field ($k$-RAT). When averaging
the polarized emission across the outer versus inner thirds of the disk, we
find that the polarization arises primarily from the SW third. We perform
synthetic observations assuming grain alignment via both $k$-RAT and $B$-RAT.
Both models produce polarization fractions close to our observed value when the
emission is averaged across the entire disk. When we average the models in the
inner versus outer thirds of the disk, we find that $k$-RAT is the likely
mechanism producing the polarized emission in $\beta$ Pic. A comparison of
timescales relevant to grain alignment also yields the same conclusion. For
dust grains with realistic aspect ratios (i.e., $s > 1.1$), our models imply
low grain-alignment efficiencies.
