  By using the GRACE-Loop system, we calculate the full $\mathcal{O}(\alpha)$
electroweak radiative corrections to the process $e^+e^- \rightarrow e^+e^-
\gamma$, which is important for future investigations at the International
Linear Collider (ILC). With the GRACE-Loop system, the calculations are checked
numerically by three consistency tests: ultraviolet finiteness, infrared
finiteness, and gauge-parameter independence. The results show good numerical
stability when quadruple precision is used. In the phenomenological results, we
find that the electroweak corrections to the total cross section range from
$\sim -4\%$ to $\sim -21\%$ when $\sqrt{s}$ varies from $250$ GeV to $1$ TeV.
The corrections also significantly affect the differential cross sections,
which are a function of the invariant masses and angles and the final-particle
energies. Such corrections will play an important role for the high-precision
program at the ILC.
