  An excellent candidate molecule for the measurement of the electron's
electric dipole moment (eEDM) is thorium monofluoride (ThF$^+$) because the
eEDM-sensitive state, $^3\Delta_1$, is the electronic ground state, and thus is
immune to decoherence from spontaneous decay. We perform spectroscopy on
$X\,^3\Delta_1$ to extract three spectroscopic constants crucial to the eEDM
experiment: the hyperfine coupling constant, the molecular frame electric
dipole moment, and the magnetic $g$-factor. To understand the impact of thermal
blackbody radiation on the vibrational ground state, we study the lifetime of
the first excited vibrational manifold of $X\,^3\Delta_1$. We perform ab initio
calculations, compare them to our results, and discuss prospects for using
ThF$^+$ in a new eEDM experiment at JILA.
