  We consider the problem of minimizing the Willmore energy in the class of
conformal immersions of a given closed, genus p Riemann surface into R^n for
n=3,4. We prove existence of a smooth minimizer, provided that the infimum is
below a certain bound ${\cal W}(n,p)$. For tori in R^3 we have explicitely
${\cal W}(3,1) = 8\pi$.
