I would use Harriet Hargrave's invisible machine applique method with freezer paper. In this case the freezer paper is ironed to the back of the fabric. Edges are turned and glued (I prefer using Elmer's white school paste instead of glue stick) and, when perfect, pressed. Pin the circle in place and machine applique with a narrow hem stitch (I usually use YLI .003 monofilament but fine silk or regular thread will be fairly invisible also). To remove the freezer paper, I run a wet finger around the edge to dampen the glue, then carefully remove the paper. (It looks like you already have the background removed; if not, you first have to make a slit in the background fabric and then remove it leaving a 1/4" to 1/2" seam allowance.)

Using freezer paper in this way ensures that the bias edges of the circle won't stretch as you sew.