  We present results of the first experiment to probe for the electric dipole
moment (EDM) of the electron using an $\Omega$-doublet state in a polar
molecule. If the molecule is both massive and has a large molecular-fixed frame
dipole moment, then the $\Omega$-doublet states have the potential to greatly
increase the sensitivity of experiments searching for the EDM while also
allowing for new methods of systematic error rejection. Here, we use the
metastable $a(1)^3\Sigma^+$ state of lead monoxide (PbO) to probe for the
electron EDM. Our best fit for the electron EDM of $d_e =
(-4.4\pm9.5_\text{stat}\pm1.8_\text{syst})\times10^{-27}\ecm$ allows us to
place an upper limit on the magnitude of the EDM of
$|d_e|<1.7\times10^{-26}\ecm$ (90% confidence). While this is less stringent
than limits from other, previous experiments, our work emphasizes the
systematic error rejection properties associated with the $\Omega$-doublet
level structure. The results should inform the work of other, ongoing
experiments that use molecules with analogous level structure.
