We discuss the relation between infinite distance andvanishing gap on the conformal manifold of unitary two-dimensionalconformal field theories (CFTs) with a normalizable conformallyinvariant vacuum. In particular we prove that any limit of vanishinggap must be located at infinite distance measured with respect to theZamolodchikov metric. We further quantify the approach to this limitin terms of exponential decay in certain operator dimensions anddeduce both upper and lower bounds on the decay rate. We also describean emergent sigma model at large radius in the limit. As a corollaryto our CFT results, we establish a part of the Distance Conjectureabout gravitational theories in three-dimensional anti-de Sitterspace, regarding the emergence of exponentially light particles on thebulk moduli space.