Let g be a positive Einstein metric on the four-sphere S^4 . If its Yamabe constant Y (S^4,[g]) = R_g \sqrt{V_g} is greater than 8¢å2¦Ð¡Ý ¦Å_0 , we show that, up to rescaling, g is isometric to the standard round metric g_S , where ¦Å_0 > 0 is a universal positive constant independent of g . This is a generalization of Gursky¡Çs gap theorem, and a partial affirmative answer to the conjecture : Any positive Einstein metric on S^4 is isometric to g_S (up to rescaling). This is a joint work with Hisaaki Endo (Tokyo Tech.) and Harish Seshadri (Indian Inst. of Sci., Bangalore).