High-resolution and high signal-to-noise ratio echelle observations made by the Goddard High Resolution Spectrograph (GHRS), aboard the Hubble Space Telescope, of molecular (CO)-C-12-O-16 and (CO)-C-13-O-16 in the absorbing gas toward zeta Oph are reported. A statistical equilibrium model, incorporating a different way of dealing with the radiation field in the interstellar clouds, is presented. Based upon previous observations, two velocity components of the gas at V(LSR) = -0.79 and 0.39 km s-1, corresponding to V. = -14.6 and -13.4 km s-1, together with three different sets of velocity dispersions are adopted in our models. The ratios of the computed emergent intensities (I(em)) resulting from the emission lines in the lowest transitions (J = 1 --> 0, 2 --> 1, 3 --> 2) of ground state of CO are presented. The ratios for the two velocity components, I(em)(1-0)1/I(em)(1-0)2, and I(em)(2-1)1/I(em)(1-0)1 are comparable to the observed corrected antenna temperature ratios. From the solutions to the statistical equilibrium equations including the six lowest rotational transitions of CO and from multiple Voigt profile fits, the observed (CO)-C-12 and (CO)-C-13 UV absorption features are successfully reproduced. Profile fitting and double cloud curve-of-growth analysis using the GHRS and Copernicus data indicate that the abundance of CO is (1.8 + 0.21 x 10(15) cm-2. The most recently measured wavelength of the (6-0) (CO)-C-13 band by Haridass & Huber [R(0) = 1370.616 angstrom] shows that the tentative (CO)-C-13 features in the GHRS data are unquestionably identified. If there is no blending from any other unknown interstellar species, the isotope ratio of (CO)-C-12/(CO)-C-13 is approximately 82 +/- 25. This result is close to the terrestrial value of 90 and implies that the CO is not substantially fractionated, and (CO)-C-12-O-16 experiences little preferential self-shielding through the selective isotopic photodissociation process. This conclusion differs strongly with that of Sheffer et al. Other implications of the results are discussed.