  Neutron stars are sensitive laboratories for testing general relativity,
especially when considering deviations where velocities are relativistic and
gravitational fields are strong. One such deviation is described by dynamical,
Chern-Simons modified gravity, where the Einstein-Hilbert action is modified
through the addition of the gravitational parity-violating Pontryagin density
coupled to a field. This four-dimensional effective theory arises naturally
both in perturbative and non-perturbative string theory, loop quantum gravity,
and generic effective field theory expansions. We calculate here Chern-Simons
modifications to the properties and gravitational fields of slowly spinning
neutron stars. We find that the Chern-Simons correction affects only the
gravitomagnetic sector of the metric to leading order, thus introducing
modifications to the moment of inertia but not to the mass-radius relation. We
show that an observational determination of the moment of inertia to an
accuracy of 10%, as is expected from near-future observations of the double
pulsar, will place a constraint on the Chern-Simons coupling constant of
\xi^{1/4} < 5 km, which is at least three-orders of magnitude stronger than the
previous strongest bound.
