A web material dispenser includes a housing and a pivotable roll support assembly provided therein. A first mandrel is provided at a first attachment location of the roll support assembly to support a first roll of web material. A second mandrel is provided at a second location of the roll support assembly to support a second roll of web material. The web material dispenser permits one-handed operation to replace the rolls of web material and to change positions of the mandrels within the dispenser. The roll support assembly may have a linkage arm and/or a roll support arm. Positions of the first and second mandrels relative to the dispenser's opening can be switched by pivoting the linkage arm relative to the housing and by pivoting the roll support arm relative to the linkage arm. A method for switching positions of rolls in the web material dispenser is also provided.