Kenny,
Thanks for the prompt reply. Everything that you wrote looks like great advice. After reading it I took a closer look at the seal. The plastic panels on the vertical sides of the cargo box help to hold and protect the part of the gasket that seals the sides of the cargo box to the tailgate. I decided not to remove the gasket completely because removing it from the rear edge of the cargo box floor was very easy. This part of the seal frequently detaches with normal use of the tailgate. I could easily inspect and access the groove of the seal that slips over the edge of the box floor while leaving the vertical ends of the it attached the box sides. It looked like it might be difficult to remove the cargo box side panels. The vertical part of the gasket is still very secure and seals well so I decided that I might do more harm than good if I tried to completely remove it. The gasket looks very good. It's not damaged, or cracked, or brittle. There was very little dirt and grime in the groove; far too little for that to be the problem. The gasket appears to fit properly, it just doesn't stay in place. The very little bit of "grime" that I did find was sticky, like adhesive. Since the rubber is very pliable I could easily feel that there are no metal clips integrated into the seal. I bought my Avalanche a year ago and within days I noticed that the seal didn't stay in place. I'm not the original owner so it's quite possible that the seal was originally detached by dragging something heavy out of the bed. The design of the rear edge of the cargo box and this seal could probably be better because I see how this could have easily happened. If that is what happened then I would expect to find some damage on the inside of the groove. If there is any damage I can't find it. Since there is no metal retainer integrated into the seal, perhaps there were separate metal clips that fit into the groove? If there were such clips they were lost before I got the vehicle. I don't think that there were separate clips because the paint that is covered by the gasket when it's in place doesn't have any scratches that such clips typically make. I think I'm going to follow your suggestion to use weather strip adhesive. That was recommended most on the other forums that I checked. Some indicated that 3M makes a good weather strip adhesive, part number 08008 for black. When I bought the vehicle it was obvious that the cargo box had not been used very much. My best guess is that the seal is supposed to be adhered and that somehow during manufacture the adhesive was not applied. After I get some adhesive and apply it, let's give it a few weeks of normal use and then I'll post the results.
Thanks again for your input.