We have made the first detection of circumstellar absorption lines of the (CC)-C-12-C-13 A (1)Pi(u)-X (1)Sigma(g)(+) (Phillips) system 1-0 band and the (CO)-C-12-O-16 X (1)Nu(+) first-overtone 2-0 band in the spectrum of the post-AGB star HD 56126 (IRAS 07134+1005). All current detections of circumstellar molecular absorption lines toward HD 56126 (C-12(2), (CC)-C-12-C-13, (CN)-C-12-N-14, (CN)-C-13-N-14, and (CO)-C-12-O-16) yield the same heliocentric velocity of V-CSE = 77.6 +/- 0.4 km s(-1). The C-12(2), (CC)-C-12-C-13, and (CO)-C-12-O-16 lines give rotational temperatures and integrated column densities of T-rot = 328 +/- 37 K, log N-int = 15.34 +/- 0.10 cm(-2), T-rot = 256 +/- 30 K, log N-int = 13.79 +/- 0.12 cm(-2), and T-rot = 51 +/- 37 K, log N-int = 18.12 +/- 0.13 cm(-2), respectively. The rotational temperatures are lower for molecules with a higher permanent dipole moment. Derived relative column densities ratios are C-12(2)/(CC)-C-12-C-13 = 36 +/- 13 and (CO)-C-12-O-16/(C-12(2) + (CC)-C-12-C-13) = 606 +/- 230. Combined with data from a previous paper, we find relative column densities of (CO)-C-12-O-16/((CN)-C-12-N-14 + (CN)-C-13-N-14) = 475 +/- 175 and (CN)-C-12-N-14/(CN)-C-13-N-14 = 38 +/- 2. Under chemical equilibrium conditions, (CC)-C-12-C-13 is formed twice as easily as C-12(2). The isotopic exchange reaction for C-12(2) is too slow to significantly alter the C-12(2)/(CC)-C-12-C-13 ratio, and the C-12(2) to (CC)-C-12-C-13 ratio is a good measure of half the carbon isotope ratio: C-12/C-13 = 2 x C-12(2)/(CC)-C-12-C-13 = 72 +/- 26. This is in agreement with our prediction that the isotopic exchange reaction for (CN)-C-12-N-14 is efficient and our observation in Paper III of (CN)-C-12-N-14/(CN)-C-13-N-14 = 38 +/- 2. A fit of the C-2 excitation model of van Dishoeck & Black to the relative population distribution of C-2 yields n(c) sigma/I = 3.3 +/- 1.0 x 10(-14). At r similar or equal to 10(16) cm, this translates into n(c) = 1.7 x 10(7) cm(-3) and (M)over dot similar or equal to 2.5 x 10(-4) M. yr(-1).