The reconstruction of the routes of invasions is one of the major challenges of invasion biology. The Ascomycete Mycophaerella fijiensis, causal agent of the Black Leaf Streak Disease of banana, stemming from South-East Asia, has started invading the American continent in 1972 and is now present in all American banana-producing regions. In a previous population genetics study at a global scale we suspected that the American invasion could have resulted from multiple introductions and admixture. In the present study, we aimed at statistically testing this hypothesis using Bayesian scenario choice based on Approximate Bayesian Computation and at inferring the demo-genetic characteristics of this introduction. To do so, we have confronted seven scenarios of disease introduction to previously acquired data consisting of molecular genotyping at 21 microsatellite markers of six American populations, two South-East Asian populations and one Oceanian population. We were able to firmly demonstrate a contribution of Oceania to this invasion. The admixture scenario received a high posterior probability (about 0.7). Parameter inferences suggest equilibrated contributions of Oceania and South-East Asia to the American invasion. The source and current invasive populations seem to have high effective population sizes (of the order of 104). Invasive populations went through a drastic bottleneck of the order of one hundred individuals for about 5 years (50 generations). Altogether these results are very congruent with historical data and are compatible with the assumption that the invasion started from fungal strains accidentally imported on infected banana plants brought back into Honduras from diverse origins in a collection built for the purpose of plant breeding.