Select the small ellipse and go to Edit > Copy (cmd/ctrl + C), then Edit > Paste in Front (cmd/ctrl + F).
Select the front small ellipse and the larger ellipse. In the Pathfinder palette (Window > Pathfinder), opt/alt + click on the Subtract from Shape Area button. When you hold opt/alt and click, the shapes are expanded without having to click the expand button. Through this whole step you won't see any change in the appearance of the shapes.


