Here's what I did to fix it, so you can keep your chair if something similar happens.
First you have to figure out the broken item in question is constructed.The arm is attached to the back with one bolt through the back frame. It screws into a threaded insert in the arm. The arm attaches to the seat with two bolts and two threaded inserts.
Two threaded wood inserts attach arm to seat
Next, figure out the root cause. I tried simply reattaching the arm but it didn't work; the threads seemed stripped. Further investigation revealed something else going on.
The bolts in the seat didn't extend out very far at all. As a result, they engaged only the first couple of threads in the inserts and those threads finally gave way. Either the countersunk holes in the seat weren't drilled deep enough, or the bolts weren't long enough (maybe I used the wrong bolts when I assembled this thing?). Fortunately, the bolts at the back do extend far enough and fully engage the inserts.
Bolt is too short to engage threaded inserts
Finally, evaluate repair and redesign options and pick the one that best meets your needs. I could have gone through the trouble of drilling the countersunk holes deeper. But why not just install longer bolts for a minimal cost (if you don't happen to have bolts of the right size and pitch)?
Well... that was easy.
To prevent a similar failure on the left side, I swapped those bolts too. Thread locking compound on the bolts will hopefully prevent them from backing out.
This chair has been in heavy use for five years or so. Aside from tje finish fading in a few spots, you may have noticed that the seat (is it leather or is it naugahyde? or pleather?) is cracked and coming apart. Clear packing tape is holding it together for now, but a recover and repadding is in order. My butt will thank me.