A compression ring to grip and compress body structure such as diverticulum, hemorrhoids, and tissue adjacent a hole. A resilient ring-shaped body defines a compression channel, and an elongated axially rigid gripping member extends diametrically across the through-opening. The gripping member can rest on a flange on the opposite side of the through-opening or engage with a second gripping member that extends diametrically across the through-opening from the opposite side of the ring. Or, a flexible cage structure can be disposed in the through-opening.