We prove the existence of a smooth minimizer of the Willmore energy in the class of conformal immersions of a given closed Riemann surface into IRn, n = 3, 4, if there is one conformal immersion with Willmore energy smaller than a certain bound Wn,p
depending on codimension and genus p of the Riemann surface. For tori in codimension 1, we know W3,1 = 8 .