  We study the $\Xi_c- \Xi_c'$ mixing effects in the semileptonic decays the
doubly charm baryons of $\Xi_{cc}$. We focus on the ratio of ${\cal
R}(\theta_c) \equiv {\cal B}( \Xi_{cc} \to \Xi_c' e^+ \nu_e)/ {\cal B}(
\Xi_{cc} \to \Xi_c e^+ \nu_e) $ and find that $({\cal R}(\theta_0),{\cal R}(-
\theta_0)) =(0.46 \pm 0.01,7.33 \pm 0.23)$ with $\theta_0 = 0.137\pi$, which
are in sharp contrast to ${\cal R}(0)=2.15\pm0.11$ without the mixing. The
ratio is enhanced~(suppressed) by a factor of four for a negative~(positive)
$\theta_c$. In addition, the polarization asymmetries of $\Xi_c^{(\prime)}$ are
found to be $\alpha(-\theta_0) = 0.32 ~(-0.76)$ and $\alpha(\theta_0) =
-0.82~(-0.38)$. As ${\cal R}$ and $\alpha $ are highly sensitive to $\theta_c$
and unaffected by the $W$-exchange contributions, they provide excellent
opportunities to determine $\theta_c$ in the ongoing experiments.
