Pull the sedum plants as they emerge from the soil in the early spring; grasp each plant near the base and pull it firmly from the ground, removing as much of the root material as possible.
Excavate the soil around the base of stubborn plants with a small shovel or garden hoe. Dig until the roots are exposed, then remove the plant.
Cover the bare ground with sections of landscape fabric, sheets of black plastic or pieces of newspaper. These materials block the sun, preventing the growth and development of any remaining shoots. With no foliage to feed it, any remaining root material will wither and die.
Spread a 2-inch layer of coarse-textured mulch, such as bark nuggets, gravel or small stones, over the fabric. The mulch holds the material in place and conceals the fabric, protecting it from the degrading effects of the sun.
Remove the materials the following spring. Pull any visible plants and repeat the process, if necessary.