This is just an idea and I don't even know if it would work.....but what about threading the weave onto a pre-formed hoop? It will mean spitting some of the Byzantine links a little to fit it through, but the hoop doesn't need to be a particularly heavy gauge and if you're using a reasonable sized ring to make the weave, it might hide it nicely. Like I said, just an idea and I've never tried it...... I know that I've used beading wire between weaves before to hold pearls in place and you can harden thin wire to create the necessary tension and shape....