In this paper, we consider the problem of computing shortest paths of bounded curvature amidst obstacles in the plane. More precisely, given two prescribed initial and final configurations (specifying the location and the direction of travel) and a set of obstacles in the plane, we want to compute a shortest $C^1$ path joining those two configurations, avoiding the obstacles, and with the further constraint that, on each $C^2$ piece, the radius of curvature is at least 1. In this paper, we consider the case of moderate obstacles (as introduced by Agarwal et al.) and present a polynomial-time exact algorithm to solve this problem.