The rotational spectral lines of c-C$_3$H$_2$ and two kinds of the $^{13}$C isotopic species, c-$^{13}$CCCH$_2$ ($C_{2v}$ symmetry) and c-CC$^{13}$CH$_2$ ($C_s$ symmetry) have been observed in the 1-3 mm band toward the low-mass star-forming region L1527. We have detected 7, 3, and 6 lines of c-C$_3$H$_2$, c-$^{13}$CCCH$_2$ , and c-CC$^{13}$CH$_2$, respectively, with the Nobeyama 45 m telescope, and 34, 6, and 13 lines, respectively, with the IRAM 30 m telescope, where 7, 2, and 2 transitions, respectively, are observed with the both telescopes. With these data, we have evaluated the column densities of the normal and $^{13}$C isotopic species. The [c-C$_3$H$_2$]/[c-$^{13}$CCCH$_2$] ratio is determined to be $310pm80$, while the [c-C$_3$H$_2$]/[c-CC$^{13}$CH$_2$] ratio is determined to be $61pm11$. The [c-C$_3$H$_2$]/[c-$^{13}$CCCH$_2$] and [c-C$_3$H$_2$]/[c-CC$^{13}$CH$_2$] ratios expected from the elemental $^{12}$C/$^{13}$C ratio are 60-70 and 30-35, respectively, where the latter takes into account the statistical factor of 2 for the two equivalent carbon atoms in c-C$_3$H$_2$. Hence, this observation further confirms the dilution of the $^{13}$C species in carbon-chain molecules and their related molecules, which are thought to originate from the dilution of $^{13}$C$^+$ in the gas-phase C$^+$ due to the isotope exchange reaction: $mathrm{^{13}C^++COrightarrow{}^{13}CO+C^+}$. Moreover, the abundances of the two $^{13}$C isotopic species are different from each other. The ratio of c-$mathrm{^{13}CCCH_2}$ species relative to c-$mathrm{CC^{13}CH_2}$ is determined to be $0.20pm0.05$. If $^{13}$C were randomly substituted for the three carbon atoms, the [c-$mathrm{^{13}CCCH_2}$]/[c-$mathrm{CC^{13}CH_2}$] ratio would be 0.5. Hence, the observed ratio indicates that c-$mathrm{CC^{13}CH_2}$ exists more favorably. Possible origins of the different abundances are discussed.
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