  We have used the Green Bank Telescope (GBT) to search for the OH molecule at
several locations in the Smith Cloud, one of the most prominent of the
high-velocity clouds that surround the Milky Way. Five positions with a high HI
column density were selected as targets for individual pointings, along with a
square degree around a molecular cloud detected with the Planck telescope near
the tip of the Smith Cloud. Gas in the Galactic disk with similar values of
$N_{HI}$ has detectable OH emission. Although we found OH at velocities
consistent with the foreground Aquila molecular cloud, nothing was found at the
velocity of the Smith Cloud to an rms level of 0.7 mK (T$_b$) in a 1 km $s^1$
channel. The three positions that give the strictest limits on OH are analyzed
in detail. Their combined data imply a $5\sigma$ limit on $N(H_2) / N_{HI} \leq
0.03$ scaled by a factor dependent on the OH excitation temperature and
background continuum $T_{ex}/(T_{ex}-T_{bg})$. There is no evidence for
far-infrared emission from dust within the Smith Cloud. These results are
consistent with expectations for a low-metallicity diffuse cloud exposed to the
radiation field of the Galactic halo rather than a product of a galactic
fountain.
