  For any unitary conformal field theory in two dimensions with the central
charge $c$, we prove that, if there is a nontrivial primary operator whose
conformal dimension $\Delta$ vanishes in some limit on the conformal manifold,
the Zamolodchikov distance $t$ to the limit is infinite, the approach to this
limit is exponential $\Delta = \exp(- \alpha t +O(1) )$, and the decay rate
obeys the universal bounds $c^{-1/2} \leq \alpha \leq 1$. In the limit, we also
find that an infinite tower of primary operators emerges without a gap above
the vacuum and that the conformal field theory becomes locally a tensor product
of a sigma-model in the large radius limit and a compact theory. As a
corollary, we establish a part of the Distance Conjecture about gravitational
theories in three-dimensional anti-de Sitter space. In particular, our bounds
on $\alpha$ indicate that the emergence of exponentially light states is
inevitable as the moduli field corresponding to $t$ rolls beyond the Planck
scale along the steepest path and that this phenomenon can begin already at the
curvature scale of the bulk geometry. We also comment on implications of our
bounds for gravity in asymptotically flat spacetime by taking the flat space
limit and compare with the Sharpened Distance Conjecture.
