An improved web material dispenser that is designed to dispense web material, such as toilet tissue or the like. The dispenser is able to retain a roll containing tissue at a dispensing position until the tissue has been exhausted from the roll. The dispenser senses that the tissue is exhausted from the roll, and only then permits a reserve roll to be rotated into a dispensing position. Thus, the dispenser ensures that the tissue from each roll is used up before permitting access to a reserve roll. The invention also provides a new web material roll that utilizes a “double core”, as well as a method of making the “double core”.