We apply to the Indian Ocean a novel technique to estimate the distribution, total mass, and net air-sea flux of anthropogenic carbon. Chlorofluorocarbon data are used to constrain distributions of transit times from the surface to the interior that are constructed to accommodate a range of mixing scenarios, from no mixing (pure bulk advection) to strong mixing. The transit time distributions are then used to propagate to the interior the surface water history of anthropogenic carbon estimated in a way that includes temporal variation in CO2 air-sea disequilibrium. By allowing for mixing in transport and for variable air-sea disequilibrium, we remove two sources of positive bias common in other studies. We estimate that the anthropogenic carbon mass in the Indian Ocean was 14.3--20.5 Gt in 2000, and the net air-sea flux was 0.26--0.36 Gt/yr. The upper bound of this range, the no-mixing limit, generally coincides with previous studies, while the lower bound, the strong-mixing limit, is significantly below previous studies.