The concept of unreliable failure detector was introduced by Chandra and Toueg as a mechanism that provides information about process failures. This mechanism has been used to solve different problems in asynchronous systems, in particular the Consensus problem.In this paper, we present a new class of unreliable failure detectors, which we call Eventually Consistent and denote by <>C. This class adds to the failure detection capabilities of other classes an eventual leader election capability. This capability allows all correct processes to eventually choose the same correct process as leader. We study the relationship between <>C and other classes of failure detectors. We also propose an efficient algorithm to transform <>C into <>P. Finally, to show the power of this new class of failure detectors, we present a Consensus algorithm based on <>C. This algorithm successfully exploits the leader election capability of the failure detector, and performs better in number of rounds than all the previously proposed algorithms for failure detectors with eventual accuracy.