In late spring, when the privet is actively growing, use a chain saw or garden loppers, depending on the size of the branches of the privet, to cut down the privet to 18 to 24 inches from the ground. Cut stumps off horizontally and keep some leaves attached to the bottom trunk and branches.
Immediately apply a solution of 25 percent glyphosate, which is the active ingredient in Roundup, to the bare stump ends before they have a chance to heal. Also, thoroughly soak the remaining leaves. Purchase a concentrated version of glyphosate to mix the right solution. Ready-mixed solutions purchased off the shelf may be as low as a 2 percent solution, which will not kill the plant. The plant must be dead before attempting removal or it will regrow from any viable root sections left behind.
If the stumps are larger than 1 inch in diameter, use a power drill to drill a hole into the end of each stump to a depth of 2 inches. Fill the holes with undiluted glyphosate concentrate and seal with a softened piece of paraffin wax. If stumps are sufficiently large, you can drill holes at a downward angle in the sides of the stump, fill with glyphosate solution and seal with paraffin wax in addition to drilling the hole in the top of the stump.
After 6 weeks, you can tell whether the stump is dead if there is no regrowth. Using a sharp ax or shovel, depending on the size of the dead privet, begin chopping around the root base of the privet at a distance close enough to separate the stump from the side roots. Using a shovel, dig the soil away from the stump as you chop away the roots until the stump begins to break away from the ground. Work downward at an angle toward the middle of the stump, chopping and digging, until all the roots are released from the stump, and pry or pull the stump from the ground. After removal, if any small sprouts are seen from leftover seeds, hand pull or treat with glyphosate solution.