Context.Carbon molecules and their¹³C-isotopologues can be used to determine the¹²C/¹³C abundance ratios in stellar and interstellar objects. C₃is a pure carbon chain molecule found in star-forming regions and in stellar shells of carbon-rich late-type stars. Latest laboratory data of¹³C-isotopologues of C₃allow a selective search for the mono-substituted species¹³CCC and C¹³CC based on accurate ro-vibrational frequencies.Aims.We aim to provide the first detection of the¹³C-isotopologues¹³CCC and C¹³CC in space and to derive the¹²C/¹³C ratio of interstellar gas in the massive star-forming region SgrB2(M) near the Galactic Center.Methods.We used the heterodyne receivers GREAT and upGREAT on board SOFIA to search for the ro-vibrational transitionsQ(2) andQ(4) of¹³CCC and C¹³CC at 1.9 THz along the line of sight towards SgrB2(M). In addition, to determine the local excitation temperature, we analyzed data from nine ro-vibrational transitions of the main isotopologue CCC in the frequency range between 1.6 and 1.9 THz, which were taken from theHerschelScience Data Archive.Results.We report the first detection of the isotopologues¹³CCC and C¹³CC. For both species, the ro-vibrational absorption linesQ(2) andQ(4) have been identified, primarily arising from the warm gas physically associated with the strong continuum source, SgrB2(M). From the available CCC ro-vibrational transitions, we derived a gas excitation temperature ofT_ex= 44.4^+4.7_−3.9K, and a total column density ofN(CCC) = 3.88^+0.39_−0.35× 10¹⁵cm⁻². Assuming the excitation temperatures of C¹³CC and¹³CCC to be the same as for CCC, we obtained column densities of the¹³C-isotopologues ofN(C¹³CC) = 2.1^+0.9_−0.6× 10¹⁴cm⁻²andN(¹³CCC) = 2.4^+1.2_−0.8× 10¹⁴cm⁻². The derived¹²C/¹³C abundance ratio in the C₃molecules is 20.5 ± 4.2, which is in agreement with the elemental ratio of 20, typically observed in SgrB2(M). However, we find theN(¹³CCC)/N(C¹³CC) ratio to be 1.2 ± 0.1, which is shifted from the statistically expected value of two. We propose that the discrepant abundance ratio arises due to the lower zero-point energy of C¹³CC, which makes position-exchange reaction converting¹³CCC to C¹³CC energetically favorable.