We have measured the isochoric heat capacity C(V) of carbon dioxide in both the one-phase and two-phase regions near the critical point. Temperatures varied from 280 K to 355 K, and densities varied from 461.2 kg.m-3 to 519.2 kg.m-3. The measurements were performed in a high-temperature adiabatic calorimeter used earlier for the determination of the isochoric heat capacity of several liquids and liquid mixtures near the critical points.(1,2) The experiments were performed at eight isochores : (461.2, 464.5, 467.5, 470.1, 477.7, 499.0, 509.8, and 519.2) kg.m-3 in the neighbourhood of the critical point and the coexistence curve. Uncertainties of the heat-capacity measurements are estimated to be less than 1.2 per cent far from T(c) and 2.5 per cent in the immediate vicinity of the critical point. The uncertainty in temperature was less than +/- 10 mK. Comparisons are made with values calculated from different scaling equation of state and with earlier measurements. The experimental results have been interpreted in terms of the modern theory of critical phenomena. The asymptotic and non-asymptotic critical behaviour of C(V) has been discussed. The critical exponents and critical amplitudes of C(V) along the critical isochore and DELTArho on the coexistence curve are derived. We have analysed the two-phase massic heat capacities of carbon dioxide to obtain detailed information about the behaviour of (partial derivative 2 p/partial derivative T2) and (partial derivative 2 mu/partial derivative T2) in the critical region.