Cut a T shape into the trunk of the weeping mulberry tree, 4 to 6 feet from the ground, with a grafting knife. Cut the horizontal slice 1 inch long and the cross section 1/3 of the diameter of the trunk.
Cut a bud from the scion 1½ inches long, scooping the bud and wood behind it in one motion with a budding knife.
Peel the bark away from the T cut to expose the trunk underneath. Tuck the bud scoop into the T cut with both newly exposed softwood sides against each other. Allow the bud to stick out from the T cut.
Wrap the trunk of the tree in grafting tape, securing the bark back over the cut section, leaving the bud poking through.
Cut the grafting tape away after 3 to 4 weeks have passed and new growth has occurred.