I didn't have to redraft the ring, but this is what I would do. I would use a compass to draw a circle on a piece of paper that is the size of the finished edge of your inner circle. This will be the inner seam line. I would then decide whether I wanted to make the ring thicker or whether I wanted to redraft the corner squares, too. If you don't want to redraft the corners, then I would draw another circle with my compass, using the same center point as the first circle, but making it the size of the circle in the pattern. This would be the other seam line. Then you could use Jackqueline's method of tracing the pattern on freezer paper, cutting on the seam lines, pressing on the fabric. Then, you can use your sewing machine to sew a 1/4" away from the seam line and cut the fabric on the sewn line, ready for sewing to your inner circle.
If you want to retain the thickness of the ring, then you would draw the second circle the correct distance (1/2"? I'm not looking at the pattern...) from the inner circle and do the same as above. This would probably look better in the end since this ring matches one of the borders. But then you would have to redraft the corners, too. For those, I would stick with the size of the square from the original pattern. To draft that pattern, I would draw a square on paper that is the finished size of one of the corner squares (so you'll end up making 4 of these). Then use my compass to draw a quarter circle centered on one of the corners of the square and the same radius as the outer circle of the ring. This is your sewing line. You can then, again, use freezer paper to put the pattern on your fabric and use your sewing machine to measure the 1/4" seam allowance. I would use my ruler to measure the seam allowance on all of the straight sides of the square before cutting out the freezer paper, but make sure to use the corner point of the square BEFORE adding the seam allowance for drawing the quarter circle.
Does that make sense? There might be easier ways, but that's how I would approach it.