Abstract: A cover mountable over a rim of a beverage container has a closed body having an interior cavity extending from an open end at one end of the body to an opposite second end. An inner surface of the sidewall of the body defines a plurality of contact locations with the rim of a beverage container to enable a single shape and size cover to be disposed on any of a number of beverage containers having different diameter rims. The inner surface of the wall of the body may have a smooth conical shape extending from a larger diameter at the first end of the body to a smaller diameter toward the second end of the body, a plurality of incremental, concentric steps of decreasing diameter between the first and second ends of the body or a plurality of ribs, each having an inner surface which tapers from a large diameter spacing at the first end of the body from adjacent ribs to a smaller diameter spacing toward the second end of the body.