A quorum-based distributed mutual exclusion protocol requires each processor in a distributed system to obtain permission from a quorum of processors before accessing a resource that cannot be concurrently shared. To prevent failed quorum members from blocking access to the resource, it is desirable to remove failed processors from quorums when failures are detected. This work addresses the problem of adapting quorums on-line, while a quorum-based mutual exclusion protocol continues to operate. To preserve the quorum intersection property that is required for mutual exclusion safety, it is necessary to coordinate changes made to the quorum data structures of different processors. A solution is given in the form of QADAPT, a decentralized algorithm that guarantees safe adaptation of quorums when processors fail. QADAPT enables any set of quorum adaptations that do not violate the quorum intersection property, and enables any set of faulty processors to be removed from quorums. QADAPT has optimal message passing cost and tolerates any number of processor (halting) failures. A distributed system model is assumed that provides only point-to-point messages with no message ordering. Results from an implementation show that the algorithm's execution rime scales well in a system containing up to fifty networked workstations. Extensions of this work include on-line adaptation of quorums that are used to maintain replica consistency in distributed databases.