Abstract: A cleaving assembly and a method for cleaving a glass body having a face at a desired angle greater than 0 degrees are disclosed. The assembly comprises a laser device for emitting a laser beam, a rotating device, and a positioning fixture. The rotating device has a head that rotates about a central axis that is orthogonal to the laser beam. The positioning fixture is operatively mounted to the head and centered axially along the central axis and is also rotatably driven by the rotating device. The positioning fixture has a tapered surface that is transverse to the central axis and that supports the glass body at a predetermined angle relative to the central axis. Rotation of the positioning fixture about the central axis when the glass body is exposed to the laser beam, cleaves the face of the glass body at the desired angle due to the glass body being supported transverse to the central axis.