Carbon burning powers scenarios that influence the fate of stars, such as the late evolutionary stages of massive stars(1) (exceeding eight solar masses) and superbursts from accreting neutron stars(2,3). It proceeds through the C-12 + C-12 fusion reactions that produce an alpha particle and neon-20 or a proton and sodium-23-that is, C-12(C-12, a) Ne-20 and C-12(C-12, p) Na-23-at temperatures greater than 0.4 x 10(9) kelvin, corresponding to astrophysical energies exceeding a megaelectronvolt, at which such nuclear reactions are more likely to occur in stars. The cross-sections(4) for those carbon fusion reactions (probabilities that are required to calculate the rate of the reactions) have hitherto not been measured at the Gamow peaks(4) below 2 megaelectronvolts because of exponential suppression arising from the Coulomb barrier. The reference rate(5) at temperatures below 1.2 x 10(9) kelvin relies on extrapolations that ignore the effects of possible low-lying resonances. Here we report the measurement of the C-12(C-12, a(0,1)) Ne-20 and C-12(C-12, p(0,1)) Na-23 reaction rates (where the subscripts 0 and 1 stand for the ground and first excited states of Ne-20 and Na-23, respectively) at centre-of-mass energies from (2.7) to 0.8 megaelectronvolts using the Trojan Horse method(6,7) and the deuteron in N-14. The cross-sections deduced exhibit several resonances that are responsible for very large increases of the reaction rate at relevant temperatures. In particular, around 5 x 10(8) kelvin, the reaction rate is boosted to more than 25 times larger than the reference value(5). This finding may have implications such as lowering the temperatures and densities(8) required for the ignition of carbon burning in massive stars and decreasing the superburst ignition depth in accreting neutron stars to reconcile observations with theoretical models(3).