We study a black hole in string theory which is described by an exact worldsheet conformal field theory (CFT). In Euclidean signature, the radial-time part of its geometry is described by a shrinking thermal cycle, resembling a cigar. It was pointed out that a certain condensate of winding string must be included in the CFT. Using the Horowitz-Polchinski effective field theory, we found that instead of ending with a smooth tip, the geometry in which a winding condensate backreacts approaches a ``puncture'' where the local radius of the thermal cycle and its derivatives vanish. The solution satisfies first-order equations, similarly to BPS configurations. We further argue that the entropy possessed by string modes that wrap the thermal cycle matches the Bekenstein-Hawking entropy. In the context of the black hole/string transition in type II superstring theory, the punctured black hole geometry has the same topology as a ``string geometry'', implying that there is no obstruction to a smooth transition from the point of view of the superconformal index, in contrast to a previous statement that was made in the literature. We comment on the Lorentzian interpretation of the solution.