Information-centric networking (ICN) is expected to be a candidate for future internet architecture, and it supports features such as multicast that improves bandwidth utilization and transmission efficiency. However, multicast itself does not provide congestion control. When multiple multicast groups coexist, multicast traffic may exhaust all network resources, and cause network congestion and packet loss. Additionally, traditional IP multicast congestion control mechanisms cannot be directly applied to ICN architecture. Therefore, it is necessary to consider an effective congestion control mechanism for ICN multicast. This paper proposes a single-rate multicast congestion control mechanism, called SRMCC. It supports router-assisted awareness of the network congestion state and congestion control message aggregation. Moreover, the fair shared rate estimation method is innovatively proposed to achieve protocol fairness. Most importantly, it adjusts the rate according to different congestion states indicated by the queue occupancy ratio. By introducing a rate selection factor, it can achieve a balance between packet loss rate and throughput. Experimental results show that our proposal outperforms other mechanisms in throughput, packet loss rate, total bandwidth utilization, and overhead, and achieves protocol fairness and better TCP friendliness.