In the context of six-dimensional conical braneworlds we consider a simple and explicit model that incorporates long distance modification of gravity and regularization of codimension-2 singularities. To resolve the conical singularities we replace the codimension-2 branes with ring-like codimension-1 branes, filling in the interiors with regular caps. The six-dimensional Planck scale in the cap is assumed to be much greater than the bulk Planck scale, which gives rise to the effect analogous to brane-induced gravity. Weak gravity on the regularized brane is studied in the case of a sharp conical bulk. We show by a linear analysis that gravity at short distances is effectively described by the four-dimensional Brans-Dicke theory, while the higher dimensional nature of gravity emerges at long distances. The linear analysis breaks down at some intermediate scale, below which fourdimensional Einstein gravity is shown to be recovered thanks to the second-order effects of the brane bending.