  An approximation to General Relativity is presented that agrees with the
Einstein field equations up to and including the fourth post-Newtonian (PN)
order. This approximation is formulated in a fully constrained scheme: all
involved equations are explicitly elliptic except the wave equation that
describes the two independent degrees of freedom of the gravitational field.
The formalism covers naturally the conformal-flat-condition (CFC) approach by
Isenberg, Wilson, and Mathews and the improved second PN-order exact approach
CFC+. For stationary configurations, like Kerr black holes, agreement with
General Relativity is achieved even through 5PN order. In addition, a
particularly interesting 2PN-exact waveless approximation is analyzed in
detail, which results from imposing more restrictive conditions. The proposed
scheme can be considered as a further development on the waveless approach
suggested by Schaefer and Gopakumar [Phys. Rev. D {\bf 69}, 021501 (2004)].
