While magnetic fields likely play an important role in driving the evolution
of protoplanetary disks through angular momentum transport, observational
evidence of magnetic fields has only been found in a small number of disks.
Although dust continuum linear polarization has been detected in an increasing
number of disks, its pattern is more consistent with that from dust scattering
than from magnetically aligned grains in the vast majority of cases. Continuum
linear polarization from dust grains aligned to a magnetic field can reveal
information about the magnetic field's direction, but not its strength. On the
other hand, observations of circular polarization in molecular lines produced
by Zeeman splitting offer a direct measure of the line-of-sight magnetic field
strength in disks. We present upper limits on the net toroidal and vertical
magnetic field strengths in the protoplanetary disk AS 209 derived from Zeeman
splitting observations of the CN 2-1 line. The 3$\sigma$ upper limit on the net
line-of-sight magnetic field strength in AS 209 is 5.0 mG on the redshifted
side of the disk and 4.2 mG on the blueshifted side of the disk. Given the
disk's inclination angle, we set a 3$\sigma$ upper limit on the net toroidal
magnetic field strength of 8.7 and 7.3 mG for the red and blue sides of the
disk, respectively, and 6.2 and 5.2 mG on the net vertical magnetic field on
the red and blue sides of the disk. If magnetic disk winds are a significant
mechanism of angular momentum transport in the disk, magnetic fields of a
strength close to the upper limits would be sufficient to drive accretion at
the rate previously inferred for regions near the protostar.