Several algorithms achieving fair rate allocation for multirate multicast flows have been proposed, but most of them are complex and unstable in the sense that they require a switch to keep many states and can not prevent the fluctuation of output queue length. We propose a control-theoretic max-min fair rate allocation algorithm for multirate multicast flows with minimum rate guarantee. The proposed algorithm doesn’t achieve only inter-and intra-session fairness keeping fewer states compared to the other algorithms, but also makes the output queue stable, which means it makes the output queue length converge to a target value at steady state. Our multirate multicast algorithm includes the consolidation algorithm which improves the transient response of the closed-loop system built up by the multirate multicast algorithm. We verify through simulations that our algorithm achieves the max-min fair rate allocation and is asymptotically stable if the stability condition, which we found through the fluid model analysis, is satisfied.