  Both celestial and momentum space amplitudes in four dimensions are beset by
divergences resulting from spacetime translation and sometimes scale
invariance. In this paper we consider a (linearized) marginal deformation of
the celestial CFT for Yang-Mills theory which preserves 2D conformal invariance
but breaks both spacetime translation and scale invariance and involves a
chirally coupled massive scalar. The resulting celestial amplitudes are
completely finite (apart from the usual soft and collinear divergences) and
take the canonical CFT form. Moreover, we show they can be simply rewritten in
terms of AdS$_3$-Witten contact diagrams which evaluate to the well-known
$D$-functions, thereby forging a direct connection between flat and AdS
holography.
