  We calculate for the first time the electric dipole moment (EDM) of $^{13}$C
generated by the isovector CP-odd pion exchange nuclear force in the
$\alpha$-cluster model, which describes well the structures of low lying states
of the $^{13}$C nucleus. The linear dependence of the EDM of $^{13}$C on the
neutron EDM and the isovector CP-odd nuclear coupling is found to be
$d_{^{13}{\rm C}} = -0.33 d_n - 0.0020 \bar G_\pi^{(1)}$. The linear
enhancement factor of the CP-odd nuclear coupling is smaller than that of the
deuteron, due to the difference of the structure between the $1/2^-_1$ state
and the opposite parity ($1/2^+$) states. We clarify the role of the structure
played in the enhancement of the EDM. This result provides good guiding
principles to search for other nuclei with large enhancement factor. We also
mention the role of the EDM of $^{13}$C in determining the new physics beyond
the standard model.
