Abstract: We discuss the relation between infinite distance and vanishing gap on the conformal manifold of unitary two-dimensional conformal field theories (CFTs) with a normalizable conformally invariant vacuum. In particular we prove that any limit of vanishing gap must be located at an infinite distance measured with respect to the Zamolodchikov metric. We further quantify the approach to this limit in terms of exponential decay in certain operator dimensions and deduce both upper and lower bounds on the decay rate. We also describe an emergent sigma model at large radius in the limit. As a corollary to our CFT results, we establish a part of the Distance Conjecture about gravitational theories in three-dimensional anti-de Sitter space, regarding the emergence of exponentially light particles on the bulk moduli space.