The diameter of a network is one of the most fundamental network parameters. Being able to compute the diameter is an important problem in the analysis of large networks, and moreover this parameter has many important practical applications in real networks. As a consequence, it is natural to study this problem in a distributed system, and more specifically in a distributed system tolerant to transient faults. More specifically, we are interested in the problem to identify one of the centers of graph. Once done, we construct a minimum diameter spanning tree rooted in this centre. Of course, the challenging problem is to compute one centre of the graph. We present a uniform self-stabilizing algorithm for the minimum diameter spanning tree construction problem in the state model. Our protocol has several attractive features that makes it suitable for practical purposes. It is the first algorithm for this problem that operates under the unfair adversary (also called unfair daemon). In other words, no restriction is made on the distributed behaviour of the system. Consequently, it is the hardest adversary to deal with. Moreover, our algorithm needs only O(log n) bits of memory per process (where n is the number of processes), that improves the previous result by a factor n. These improvements are not achieved to the detriment of the convergence time, that stays reasonable with O(n2) rounds.