In this paper, the mobility and motion characteristics of a special 3-UPU parallel mechanism (PM) are studied. Using the screw theory and geometric method, the mobility of this PM is analyzed. When the moving platform and the base are not parallel, it has two rotational and one translational DOF. There is always a midperpendicular plane between the base and the moving platform. The moving platform can realize continuous rotation about any fixed axis or any fixed point on the midperpendicular plane and can have continuous translation along the normal line of the midperpendicular plane. Two numerical examples of the continuous motion are given.