The solidification path of alloy reveals the detailed relationship between the solute concentration in liquid and the temperature during the solidification process. The best and most accurate method to predict the solidification path of multicomponent/multiphase alloys is to establish proper microsegregation modeling coupled with phase diagram calculations according to the CALPHAD method. Recently, several alloy systems such as Al-Cu, Al-Mg and Cu-Mg have been developed, which have aroused the interest of many researchers. Up to now, the research about Al-Cu-Mg ternary alloy, especially containing higher Mg content, is relatively rare. The purpose of the present work is to investigate the solidification path of Al-6.32Cu-25.13Mg (mass fraction, %) ternary eutectic alloy at different cooling rates and solid back diffusion coefficients by an extended unified microsegregation model coupled with Thermo-Calc. Solidification experiments and subsequent microstructural characterization are combined with numerical calculation of solidification paths. It was shown that the cooling rates R_f had no obvious effect on the solidification path which was (L+α)→(L+α+T)→(L+α+β+T); but the solid back diffusion coefficient Φ had a great effect on the solidification path, which evolved gradually from (L+α)→(L+α+T)→(L+α+β+T) into (L+α)→(L+α+T) when Φ increased from 0 to 1. The volume fractions of primary α phase V_α, binary eutectic V_2E and ternary eutectic V_3E at each solidification path were calculated. It was shown that V_2E decreased with the increase of R_f whereas V_3E increased and V_α was almost invariant. The dependence of V_2E, V_3E and R_f were determined by linear regression analysis given as: V_2E=-2.5lgR_f+64.9, V_3E=2.5lgR_f+22.12. The increase in Φ led to increases in V_α and V_2E and decrease in V_3E. The predicted solidification paths and volume fractions of Al-6.32Cu-25.13Mg ternary eutectic alloy at different cooling rates were in good agreement with experimental results.