Set the chair on its back or side to give you access to the chair legs. Pry the chair legs gently apart, and remove the loose rung.
Wrap a piece of 100 grit sandpaper on the end of a wooden spoon, and insert the spoon end into the leg socket. Twist the spoon to ream the glue out of the socket with the sandpaper.
Scrape the old glue off the ends of the rung, using a utility knife or a sheet of 100 grit sandpaper.
Install a 5/64-inch drill bit on a power drill. Drill a hole through the chair leg, into the leg socket.
Apply carpenter's glue onto the ends of the rung, and inside the leg sockets. Reinstall the rung ends into the leg sockets. The hole you drilled allows excess glue to ooze out so the rung seats fully inside the sockets.
Wipe off the excess glue from the chair, using a rag. Tap a finishing nail into the hole you drilled in the chair leg, using a hammer. Wait 24 hours before using the chair, only as a precaution to allow the glue to fully dry.