May roundabouts be safer for cyclists than intersections? How are safe roundabouts designed? This paper tries to answer these questions on the basis of a before-after safety study of conversions of intersections to 255 single-lane roundabouts in Denmark. The before-after study accounts for long-term accident and injury trends and regression-to-the-mean effects. In order to relate safety effects for cyclists of various roundabout design features it is crucial to split the converted sites by speed limit, because safety effects for both cyclists and other road users of converting intersections to roundabouts depend heavily on speed limits on roads entering the converted sites. If speed limits are 70km/h or higher then converting intersections to roundabouts have resulted in bicycle safety improvements in Denmark. Results show that diameter and height of central islands and type of bicycle facilities at single-lane roundabouts have considerable impacts on cyclists' safety. Central island diameters of 20-40m are safer for cyclists than smaller or larger roundabouts. A central island, which middle is elevated 2m or more above the circulating lane, is safer for cyclists than single-lane roundabouts with lower central islands. Single-lane roundabouts with separate cycle paths, where cyclists must yield to motorists entering or exiting the roundabout, are safer than roundabouts with cycle lanes. Single-lane roundabouts are safer for cyclists than intersections regardless of speed limits when these roundabouts have high central islands and/or separate cycle paths.