  The paper presents the conservative dynamics of two-body point-mass systems
up to the third post-Newtonian order ($1/c^6$). The two-body dynamics is given
in terms of a higher order ADM Hamilton function which results from a third
post-Newtonian Routh functional for the total field-plus-matter system. The
applied regularization procedures, together with making use of distributional
differentiation of homogeneous functions, give unique results for the terms in
the Hamilton function apart from the coefficient of the term $(\nu
p_{i}{\pa_{i}})^2r^{-1}$. The result suggests an invalidation of the binary
point-mass model at the third post-Newtonian order.
