We consider the problem of recovering causal structure from multivariate observational data. We assume that the data arise from a linear structural equation model (SEM) in which the idiosyncratic errors are allowed to be dependent in order to capture possible latent confounding. Each SEM can be represented by a graph where vertices represent observed variables, directed edges represent direct causal effects, and bidirected edges represent dependence among error terms. Specifically, we assume that the true model corresponds to a bow-free acyclic path diagram, i.e., a graph that has at most one edge between any pair of nodes and is acyclic in the directed part. We show that when the errors are non-Gaussian, the exact causal structure encoded by such a graph, and not merely an equivalence class, can be consistently recovered from observational data. The Bow-free Acylic Non-Gaussian (BANG) method we propose for this purpose uses estimates of suitable moments, but, in contrast to previous results, does not require specifying the number of latent variables a priori. We illustrate the effectiveness of BANG in simulations and an application to an ecology data set.