  The metastable $H \ {}^3\Delta_1$ state in the thorium monoxide (ThO)
molecule is highly sensitive to the presence of a CP-violating permanent
electric dipole moment of the electron (eEDM). The magnetic dipole moment
$\mu_H$ and the molecule-fixed electric dipole moment $D_H$ of this state are
measured in preparation for a search for the eEDM. The small magnetic moment
$\mu_H = 8.5(5) \times 10^{-3} \ \mu_B$ displays the predicted cancellation of
spin and orbital contributions in a ${}^3 \Delta_1$ paramagnetic molecular
state, providing a significant advantage for the suppression of magnetic field
noise and related systematic effects in the eEDM search. In addition, the
induced electric dipole moment is shown to be fully saturated in very modest
electric fields ($<$ 10 V/cm). This feature is favorable for the suppression of
many other potential systematic errors in the ThO eEDM search experiment.
