Rising atmospheric CO<sub>2</sub> concentrations due to anthropogenic emissions induce changes in the carbonate chemistry of the oceans and, ultimately, a drop in ocean pH. This acidification process can harm calcifying organisms like coccolithophores, molluscs, echinoderms, and corals. It is expected that ocean acidification in combination with other anthropogenic stressors will cause a severe decline in coral abundance by the end of this century, with associated disastrous effects on reef ecosystems. Despite the growing importance of the topic, little progress has been made with respect to modelling the impact of acidification on coral calcification. Here we present a model for a coral polyp that simulates the carbonate system in four different compartments: the seawater, the polyp tissue, the coelenteron, and the calcifying fluid. Precipitation of calcium carbonate takes place in the metabolically controlled calcifying fluid beneath the polyp tissue. The model is adjusted to a state of activity as observed by direct microsensor measurements in the calcifying fluid. We find that a transport mechanism for bicarbonate is required to supplement carbon into the calcifying fluid because CO<sub>2</sub> diffusion alone is not sufficient to sustain the observed calcification rates. Simulated CO<sub>2</sub> perturbation experiments reveal decreasing calcification rates under elevated <i>p</i>CO<sub>2</sub> despite the strong metabolic control of the calcifying fluid. Diffusion of CO<sub>2</sub> through the tissue into the calcifying fluid increases with increasing seawater <I>p</i>CO<sub>2</sub>, leading to decreased aragonite saturation in the calcifying fluid. Our modelling study provides important insights into the complexity of the calcification process at the organism level and helps to quantify the effect of ocean acidification on corals.