To provide a method for driving a dampening roller by which a refuse stuck on a plate cylinder is removed when a place which does not directly influence on regular printing number of sheets passes through a printing unit even if a defect on printing is generated, and hickey can be prevented from being generated and an apparatus for driving it.

In a water feeding device part of the printing unit, an equal speed driving circuit 31 for rotating the dampening roller 65 for sticking dampening water on the plate cylinder 50 at approximately the same peripheral speed as the peripheral speed of the plate cylinder 50 and a reduction speed driving circuit 32 for rotating the dampening roller 65 at a slower peripheral than the peripheral speed of the plate cylinder 50 are provided and when regular printing is operated, the operation is switched to the equal speed driving circuit 31 and when a web connection time is detected, the regular printing is stopped and it is switched to the reduction speed driving circuit 32 to transfer the refuse stuck on the plate cylinder 50 to the dampening roller 65 side and to remove it.