  We report measurements of the photon beam asymmetry $\Sigma$ for the reaction
$\vec{\gamma} p\to K^+\Sigma^0$(1193) using the GlueX spectrometer in Hall D at
Jefferson Lab. Data were collected using a linearly polarized photon beam in
the energy range of 8.2-8.8 GeV incident on a liquid hydrogen target. The beam
asymmetry $\Sigma$ was measured as a function of the Mandelstam variable $t$,
and a single value of $\Sigma$ was extracted for events produced in the
$u$-channel. These are the first exclusive measurements of the photon beam
asymmetry $\Sigma$ for the reaction in this energy range. For the $t$-channel,
the measured beam asymmetry is close to unity over the $t$-range studied,
$-t=(0.1-1.4)~$(GeV/$c$)$^{2}$, with an average value of $\Sigma = 1.00\pm
0.05$. This agrees with theoretical models that describe the reaction via the
natural-parity exchange of the $K^{*}$(892) Regge trajectory. A value of
$\Sigma = 0.41 \pm 0.09$ is obtained for the $u$-channel integrated up to
$-u=2.0$~(GeV/$c$)$^{2}$.
