  Infrared divergences in perturbative gravitational scattering amplitudes have
been recently argued to be governed by the two-point function of the
supertranslation Goldstone mode on the celestial sphere. We show that the form
of this celestial two-point function simply derives from an effective action
that also controls infrared divergences in the symplectic structure of General
Relativity with asymptotically flat boundary conditions. This effective action
finds its natural place in a path integral formulation of a celestial conformal
field theory, as we illustrate by re-deriving the infrared soft factors in
terms of celestial correlators. Our analysis relies on a well-posed action
principle close to spatial infinity introduced by Comp\`ere and Dehouck.
