To guarantee the delivery of real-time messages before their deadline, a real-time channel or connection must be established before the transmission of any real-time messages. During this channel establishment phase, one must first select a route between the source and destination of this channel and then reserve sufficient resources along this route so that the end-to-end delay over the selected route may not exceed the user-specified delay bound.We propose an efficient distributed route-selection scheme that is guaranteed to find a "qualified" route, if any, satisfying the performance requirement of the requested channel without compromising any of the existing guarantees. The proposed scheme can also eliminate the common reliability /performance bottleneck of a centralized route-selection scheme, while improving efficiency over the centralized and other distributed schemes. Simulation results are presented to demonstrate the effectiveness of the proposed distributed route-selection scheme.