  This paper describes the results of the most recent measurement of the
permanent electric dipole moment (EDM) of neutral $^{199}$Hg atoms. Fused
silica vapor cells containing enriched $^{199}$Hg are arranged in a stack in a
common magnetic field. Optical pumping is used to spin-polarize the atoms
orthogonal to the applied magnetic field, and the Faraday rotation of
near-resonant light is observed to determine an electric-field-induced
perturbation to the Larmor precession frequency. Our results for this frequency
shift are consistent with zero; we find the corresponding $^{199}$Hg EDM
$d_{Hg} = (2.20 \pm 2.75_{stat} \pm 1.48_{syst}) \times 10^{-30} e\cdot
\text{cm}$. We use this result to place a new upper limit on the $^{199}$Hg EDM
$|d_{Hg}| < 7.4\times 10^{-30} e\cdot \text{cm}$ (95\% C.L.), improving our
previous limit by a factor of 4. We also discuss the implications of this
result for various $CP$-violating observables as they relate to theories of
physics beyond the standard model.
