C-12 and C-13 abundances have been measured in 11 bright giant members of the globular cluster omega Centauri via observations of the first-overtone CO bands near 2.3 mum. The stars in this sample were selected to span a substantial fraction of the range of iron abundances found in this cluster. In addition, the sample covers a range of [O/Fe], [Na/Fe], and [Al/Fe] abundance ratios derived in previous studies. In all omega Cen giants the C-12/C-13 abundance ratio is found to be quite low, indicating deep mixing in these red giants. The mean value for the entire sample is [C-12/C-13] = 4.3 +/- 0.4(sigma = 1.3), with nine stars equal, within the errors, to the equilibrium ratio C-12/C-13 = 3.5 and two stars having slightly higher values. There is no correlation between the C-12/C-13 and the abundance of iron. In addition, no correlation of C-12/C-13 with [C-12/Fe] is found ( all giants are deeply mixed), although the derived abundances of [C-12/Fe] show a positive correlation with [O/Fe] and an anticorrelation with [Na/Fe] (with the oxygen and sodium abundances taken from previous studies in the literature). A comparison of the isotopic carbon ratios in omega Cen with those from other globular clusters (M4, M71, NGC 6752, and 47 Tuc) and with literature oxygen abundances, may reveal a slight trend of decreasing C-12/C-13 ratios with decreasing [O/Fe] in the entire globular cluster sample of red giants. A comparison between C-12/C-13 and both [Na/Fe] and [Al/Fe], however, reveals no trend.