  The Willmore energy of a closed surface in R^n is the integral of its squared
mean curvature, and is invariant uner M\"obius transformations of R^n. We show
that any torus in R^3 with energy at most $8 \pi-delta$ has a representative
under the M\"obius action, for which the induced metric and a conformal metric
of constant (zero) curvature are uniformly equivalent, with constants depending
only on $delta>0$. An analogous estimate is also obtained for surfaces of fixed
genus $p \geq 1$ in R^3 or R^4, assuming suitable energy bounds which are sharp
for n=3. Moreover the conformal type is controlled in terms of the energy
bounds.
