The context of this work is the well-studied dissemination of information in large-scale distributed networks through pairwise interactions. This problem, originally called rumor mongering, and then rumor spreading, has mainly been investigated in the synchronous model. This model relies on the assumption that all the nodes of the network act in synchrony; that is, at each round of the protocol, each node is allowed to contact a random neighbor. In this paper, we drop this assumption under the argument that it is not realistic in large-scale systems. We, thus, consider the asynchronous variant, with which, at random times, nodes successively interact by pairs, exchanging their information on the rumor. In a previous paper, we performed a study of the total number of interactions needed for all the nodes of the network to discover the rumor. Although most of the existing results involve huge constants that do not allow us to compare different protocols, we provided a thorough analysis of the distribution of this total number of interactions together with its asymptotic behavior. In this paper, we extend this discrete-time analysis by solving a conjecture proposed previously, and we consider the continuous-time case, in which a Poisson process is associated to each node to determine the instants at which interactions occur. The rumor-spreading time is, thus, more realistic because it is the real time needed for all the nodes of the network to discover the rumor. Once again, as most of the existing results involve huge constants, we provide tight bound and equivalent of the complementary distribution of the rumor-spreading time. We also give the exact asymptotic behavior of the complementary distribution of the rumor-spreading time around its expected value when the number of nodes tends to infinity.