  We consider the class of all conformal mappings from a compact Riemann
surface into the threedimensional or fourdimensional Euclidean space. A
sequence in this class with bounded Willmore functional is shown to have a
sequence of conformal transformations of the target space, such that a
subsequence of the transformed sequence converges. This implies that there
exists a smooth conformal mapping, which minimizes the Willmore functional in
this class. For this purpose we extend the quaternionic function theory of
Pedit and Pinkall to square integrable Hopf fields. In particular, we proof the
Pluecker formula for such Hopf fields.
