Heat a small bowl of vinegar in the microwave for 10-20 seconds. It should be warm, not hot. The vinegar will help remove the old glue on the chair rung.
Soak the end of a cotton swab in the vinegar. Dab the cotton swap at each end of the loose rung, as far into the joint as you can go.
Scrape away as much of the glue as you can with a dull knife, being careful not to shave too much of the wood off.
Apply fresh wood glue to each end.
Tie a rope around the chair legs and insert a wooden dowel or other short stick in the knot. Next, turn the dowel to tighten the rope to create the pressure needed for the loose rung to stay in its proper place.
Leave the dowel tucked behind the chair rung to keep the rope taut and maintain pressure. Let the glue dry for 24 hours before removing the dowel and rope.