If a solution (M,g(t)) of Ricci flow develops a local singularity at a finite time T, there is a proper subset S of M on which the curvature becomes infinite as time approaches T. Existing approaches to Ricci-flow-with-surgery, due to Hamilton and Perelman, require one to modify the solution in a small neighborhood of S by gluing in a highly curved but nonetheless nonsingular solution. This must be done with careful regard to various surgery parameters in order to preserve critical a priori estimates. In case the local singularity is a rotationally-symmetric neckpinch (in any dimension $n>2$), we can now restart Ricci flow directly from the singular limit g(T), without performing an intervening surgery or making ad hoc choices. We show that any complete smooth forward evolution from g(T) is necessarily compact and has a unique asymptotic profile as it emerges from the singularity, which we describe. (This is joint work with Sigurd Angenent and Cristina Caputo.)