We investigate point singularities of Willmore surfaces, which for example appear as blowups of the Willmore flow near singularities, and prove that closed Willmore surfaces with one unit density point singularity are smooth in codimension one. As applications we get in codimension one that the Willmore flow of spheres with energy less than 8π exists for all time and converges to a round sphere and further that the set of Willmore tori with energy less than 8π − δ is compact up to M¨bius transformations. o 1. Introduction For an immersed closed surface f : ...