Fortunately, there's a garage door supply shop in the next town, so I went and got a replacement. But replacing the spring means you have to wind it up with these large metal bars. So imagine the largest spring in your house bring tightened up a couple of inches from your jugular: if this thing snaps, well, you're gonna have a bad time.
Fortunately, with a little help from my son, everything went fine. Before winding, I spray-painted a line across the coil; after winding, you can see it went through six full revolutions.