Purchase young tarragon plants that have a minimum of three green shoots. Young plants divided from an older, parent plant tend to be hardier and thrive when compared to those grown from seeds.
Fill a 12-by-12-inch pot with a sandy, loam soil mixture. This soil type is ideal for French tarragon because the plant prefers a well-drained soil bordering on dry conditions. Dig a hole in the center of the pot, about 4 inches deep. Carefully lift the young tarragon plant from its holding container and place it into the hole.
Set the tarragon pots outside just after the final frost of the growing season. Water plants when the soil feels dry to the touch. Water until the soil feels moist, but do not allow water to pool in the pot. Excessive water can damage tarragon roots and leaves.
Pick young tarragon leaves from the plants throughout the summer. In mid-autumn, cut back the plants to help encourage dormancy. Most tarragon plants require a minimum of a two-month dormant period to thrive. In warm climates, bring tarragon plants into a cool location, like a garage. Keep in a dark corner until after the final frost.
Dig up tarragon plants every two years. Use your hands to divide the plant into three or four specimens. Replant the divided specimens in new pots. This helps prevent French tarragon from becoming too large for its container.