We consider the problem of solving consensus using deterministic algorithms in a synchronous dynamic network with unreliable, directional point-to-point links, which are under the control of a message adversary. In contrast to a large body of existing work that focuses on oblivious message adversaries where the communication graphs are picked from a predefined set, we consider message adversaries where guarantees about stable periods that occur only eventually can be expressed. We reveal to what extent such eventual stability is necessary and sufficient, that is, we present the shortest period of stability that permits solving consensus, a result that should prove quite useful in systems that exhibit erratic boot-up phases or recover after repeatedly occurring, massive transient faults. Contrary to the case of longer stability periods, where we show how standard algorithmic techniques for solving consensus can be employed, the short-lived nature of the stability phase forces us to use more unusual algorithmic methods that avoid waiting explicitly for the stability period to occur.