www.rosebrand.com has adhesive black duvetyne tape in various widths. Next to black velvet or felt, this is about as light-absolbing as you can get for a border.
For my screen, I used 1x4 poplar wood (very straight, but expensive), inside edges routed with a 45 degree bit to reduce shadow casting, then wrapped with 16oz black velvet. To attach the velvet, I used spray adhesive on the front side, and my electric staple gun on the backside of the border. It was then screwed from behind to my screen frame.
The results are most impressive!
HTH,