  We prove that if a simply connected non-conformally flat conformal Lorentzian
manifold $(M,c)$ admits an essential transitive group of conformal
transformations, then there exists a metric $g\in c$ such that $(M,g)$ is a
complete homogeneous plane wave. We also prove that the group of conformal
transformations of a non-conformally flat simply connected homogeneous plane
wave $(M,g)$ consists of homotheties, hence it is a 1-dimensional extension of
the group of isometries.
