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Our Bissell 8910 brush belt now stays in place.
The answer to my "keeper" problem was located in Peggy's button box. 
This is the black brush arm. It rests
tightly against the sweeper frame.
Inside the black brush arm with two buttons cemented in place.
If you try this, you will have to estimate how much room you want to leave between the toothed pully belt and the black brush arm. You want enough space to allow a slightly wider belt than the original but not enough space to allow the belt to run off the pulley.  

Reassembly is a little tricky. With the belt in place on the pulley, hold the brush up while you slip the black brush arm onto the brush shaft, then with the brush and the arm held up, press the back of the arm down into its resting place, then let the brush drop into its normal runnning space.  Then replace the red holder that you removed at first, back in place and screw it down. 

I suggest that you use the Bissell User Guide, which you can download free from several sources as a PDF for more information about disassembly and reassembly.  Look on pages 18 and 19 of the PDF.

Below: this is how it works to hold the belt in place.
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NOTICE:  This site is not a recommendation that you try doing what I have done. If you have trouble with your carpet cleaner that you can't resolve, take it to a qualified repair shop or purchase a new cleaner.  I prefer to 'tinker' with my own devices and I do not recommend that you do the same. This site is posted only to show what I have done to my own property.
Two plastic buttons were exactly the right size.
Use a good cement to hold them in place.