The concept of unreliable failure detector was introduced by Chandra and Toueg [2] as a mechanism that provides information about process failures. Depending on the properties the failure detectors guarantee, they proposed a taxonomy of failure detectors. It has been shown that one of the classes of this taxonomy, namely Eventually Strong (¿Ë), is the weakest class allowing to solve the Consensus problem. In this paper, we present a new algorithm implementing ¿Ë. Our algorithm guarantees that eventually all the correct processes agree on a common correct process. This property trivially allows us to provide the accuracy and completeness properties required by ¿Ë. We show, then, that our algorithm is better than any other proposed implementation of ¿Ë in terms of the number of messages and the total amount of information periodically sent. In particular, previous algorithms require to periodically exchange at least a quadratic amount of information, while ours only requires ÇÂ...