The double triangle algebra(DTA) associated to an ADE graph is considered. A description of its bialgebra structure based on a reconstruction approach is given. This approach takes as initial data the representation theory of the DTA as given by Ocneanu's cell calculus. It is also proved that the resulting DTA has the structure of a weak *-Hopf algebra. As an illustrative example, the case of the graph A3 is described in detail.