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$^{+}$.
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