Remove the screw plug and screws from the spot where the arms meet the chair back or from the underside of the chair. Pull the arm rests up and unscrew or remove the spindles or support bars.
Measure the diameter of the spindles or supports and purchase dowel screws at least half that size.
Fit an electric drill with a drill bit one size smaller than the dowel screws and drill pilot holes into the bottom of the spindles or supports and in the spots on the chair seat where the spindles were removed.
Fit an electric drill with a dowel screw driver bit and install the screws in the ends of the spindles or supports.
Screw the spindles or supports into the pilot holes in the chair seat. Replace the arm rests and the screws from the chair back.
Insert the tip of a bottle of wood glue into the gaps between the chair arm spindles or support bars.
Squeeze a drop of glue into the gap.
Allow the glue to dry for at least two hours, then test the arms. Add a bit more glue if necessary.