  We have carried out observations of CCH and its two $^{13}$C isotopologues,
$^{13}$CCH and C$^{13}$CH, in the 84 - 88 GHz band toward two starless cores,
L1521B and L134N (L183), using the Nobeyama 45 m radio telescope. We have
detected C$^{13}$CH with a signal-to-noise (S/N) ratio of 4, whereas no line of
$^{13}$CCH was detected in either the dark clouds. The column densities of the
normal species were derived to be ($1.66 \pm 0.18$)$\times 10^{14}$ cm$^{-2}$
and ($7.3 \pm 0.9$)$\times 10^{13}$ cm$^{-2}$ ($1 \sigma$) in L1521B and L134N,
respectively. The column density ratios of $N$(C$^{13}$CH)/$N$($^{13}$CCH) were
calculated to be $>1.1$ and $>1.4$ in L1521B and L134N, respectively. The
characteristic that $^{13}$CCH is less abundant than C$^{13}$CH is likely
common for dark clouds. Moreover, we find that the $^{12}$C/$^{13}$C ratios of
CCH are much higher than those of HC$_{3}$N in L1521B by more than a factor of
2, as well as in Taurus Molecular Cloud-1 (TMC-1). In L134N, the differences in
the $^{12}$C/$^{13}$C ratios between CCH and HC$_{3}$N seem to be smaller than
those in L1521B and TMC-1. We discuss the origins of the $^{13}$C isotopic
fractionation of CCH and investigate possible routes that cause the
significantly high $^{12}$C/$^{13}$C ratio of CCH especially in young dark
clouds, with the help of chemical simulations. The high $^{12}$C/$^{13}$C
ratios of CCH seem to be caused by reactions between hydrocarbons (e.g., CCH,
C$_{2}$H$_{2}$, $l,c$-C$_{3}$H) and C$^{+}$.
