  We report results of an experimental search for the intrinsic Electric Dipole
Moment (EDM) of the electron using a solid-state technique. The experiment
employs a paramagnetic, insulating gadolinium gallium garnet (GGG) that has a
large magnetic response at low temperatures. The presence of the eEDM would
lead to a small but non-zero magnetization as the GGG sample is subject to a
strong electric field. We search for the resulting Stark-induced magnetization
with a sensitive magnetometer. Recent progress on the suppression of several
sources of background allows the experiment to run free of spurious signals at
the level of the statistical uncertainties. We report our first limit on the
eEDM of $(-5.57 \pm 7.98 \pm 0.12)\times$10$^{-25}$e$\cdot$cm with 5 days of
data averaging.
