Combine equal amounts of moist garden soil, coarse sand and peat moss to produce a succulent medium. Fill a 4- to 6-inch plant pot with the soil mixture.
Cut a small branching stem from the parent euphorbia at the base where the small stem meets the larger stem, using the razor knife.
Set the cutting in a warm protected area, away from humidity. Allow the cut end of the cutting to dry out and form a callus. This helps prevent the cut end from rotting when planted.
Dig a hole deep enough so the planted portion of the stem will support the plant. Insert the cut end of the cutting into the soil. Press the soil down around the stem.
Place the newly planted cutting in a warm and sunny area. Water the cutting after one week. Watch for new growth. Water sparingly, allowing the soil to dry out completely between waterings, as the plant becomes established.