The development of reliable distributed software is simplified by the ability to assume a fail-stop failure model. We discuss the emulation of such a model in an asynchronous distributed environment. The solution we propose, called Strong-GMP, can be supported through a highly efficient protocol, and has been implemented as part of a distributed systems software project at Cornell University. Here, we focus on the precise definition of the problem, the protocol, correctness proofs and an analysis of costs. Keywords: Asynchronous computation; Fault detection; Process membership; Fault tolerance; Process group.