We consider repeated games where the number of repetitions u is unknown. The information about the uncertain duration can change during the play of the game. This is described by an uncertain duration process U that defines the probability law of the signals that players receive at each stage about the duration. To each repeated game G and uncertain duration process U is associated the U-repeated game G(U). A public uncertain duration process is one where the uncertainty about the duration is the same for all players. We establish a recursive formula for the value V_U of a repeated two-person zero-sum game G(U) with a public uncertain duration process U. We study asymptotic properties of the normalized value v_U = V_U/E(u) as the expected duration E(u) goes to infinity. We extend and unify several asymptotic results on the existence of lim v_n and lim v_ë and their equality to lim v_U. This analysis applies in particular to stochastic games and repeated games of incomplete information.