Since the multi-rate enhancements have been implemented in wireless ad hoc networks (MANETs), QoS-constrained multicast protocols for multimedia communication should be adapted to exploit them fully for using more efficiently the limited resources. To build a multicast tree with delay-guaranteed, one-hop delay and end-to-end delay must be known. The one-hop delay is the transmission time at a given link connecting two neighboring hosts, and the end-to-end delay is the time taken for a data packet from a specific source to reach the destination node. However, how to calculate the above two delays using the IEEE 802.11 MAC is still a challenging problem, because the radio channel is shared among neighbors. Further, the multi-rate enhancements make it more difficult since the neighboring relation among hosts is varied with the transmission rates. In this paper, we first propose a method to estimate the one-hop delay based on varied transmission rates by monitoring the sensed busy/idle ratio of shared channel. Then, by its aid, another method is proposed to compute the end-to-end delay. Finally, we integrate the above two methods with a typical multicast routing protocol, ODMRP, for constructing a delay-guaranteed multicast protocol. Simulation results show that the proposed method obtains more precise one-hop delay than a very recently work. Besides, the integrated protocol provides better delay guarantee than the existing protocol when the multicast traffic has delay requirement.