A surgical blade package facilitates the mounting of a surgical blade on to the forwardly-projecting cleat on a scalpel handle; and following use in the O.R., the surgical blade package facilitates stripping the used blade off the scalpel for disposal in a sharps container. The cleat is inserted beneath the rear end portion of the blade, and between the blade and the bottom wall of the blade package; and forward and upward movement of the scalpel snaps the blade on to the scalpel in a continuous movement. The opposite procedure is used to demount the used blade from the scalpel and encase it in the original blade package for disposal in a sharps container.