A closure for a container having an opening includes a base cap and an overcap that form a sealed chamber adjacent the opening that is isolated from the contents of the container. The base cap includes an outer skirt having container-engaging structure, a cylindrical well, a frangible membrane connected to the well along an inclined line of weakness and by a hinge member. The hinge member includes a pocket extending downward adjacent lower and upper terminuses of the line of weakness. The overcap includes a body having gripping structure, an inner skirt received within and rotatably connected to the well, and a cutting member depending from a lower end of the inner skirt received within the pocket such that the cutting member extends below the upper terminus. The cutting member severs the line of weakness upon substantial rotation of the overcap with respect to the base cap. A method of using the closure is also disclosed.