  Asymptotic particle states in four-dimensional celestial scattering
amplitudes are labelled by their $SL(2,\mathbb{C})$ Lorentz/conformal weights
$(h,\bar{h})$ rather than the usual energy-momentum four-vector. These boost
eigenstates involve a superposition of all energies. As such, celestial gluon
(or photon) scattering cannot obey the usual (energetically) soft theorems. In
this paper we show that tree-level celestial gluon scattering, in theories with
sufficiently soft UV behavior, instead obeys conformally soft theorems
involving $h \to 0$ or $\bar{h} \to 0$. Unlike the energetically soft theorem,
the conformally soft theorem cannot be derived from low-energy effective field
theory.
