  Asymptotically harmonic manifolds are simply connected complete Riemannian
manifolds without conjugate points such that all horospheres have the same
constant mean curvature $h$. In this article we present results for harmonic
functions on rank one asymptotically harmonic manifolds $X$ with mild curvature
boundedness conditions. Our main results are (a) the explicit calculation of
the Radon-Nykodym derivative of the visibility measures, (b) an explicit
integral representation for the solution of the Dirichlet problem at infinity
in terms of these visibility measures, and (c) a result on horospherical means
of bounded eigenfunctions implying that these eigenfunctions do not admit
non-trivial continuous extensions to the geometric compactification $\bar{X}$.
