Cut the honeysuckle stem as close to the ground as possible. Depending on the variety and age of the honeysuckle, it may be necessary to use anything from pruning shears to a saw to cut through the stem.
Apply a glyphosate herbicide directly to the cut surface of the stem immediately after you cut through it. Brush the herbicide on the stump with a paintbrush, with extra attention to where the bark meets the heartwood as this area absorbs the herbicide more efficiently than other areas.
Monitor the honeysuckle stump over the next few months for new growth. Apply more herbicide each time more growth occurs. Depending on how established the honeysuckle was, it may take several applications to fully kill the stump.