  The leading long-distance quantum correction to the Newtonian potential for
heavy spinless particles is computed in quantum gravity. The potential is
obtained directly from the sum of all graviton exchange diagrams contributing
to lowest non-trivial order to the scattering amplitude. The calculation
correctly reproduces the leading classical relativistic post-Newtonian
correction. The sign of the perturbative quantum correction would indicate
that, in the absence of a cosmological constant, quantum effects lead to a slow
increase of the gravitational coupling with distance.
