  We consider surfaces in ${\mathbb R}^3$ of type ${\mathbb S}^2$ which
minimize the Willmore functional with prescribed isoperimetric ratio. The
existence of smooth minimizers was proved by Schygulla (Archive Rational
Mechanics and Analysis, 2012). In the singular limit when the isoperimetric
ratio converges to zero, he showed convergence to a double round sphere in the
sense of varifolds. Here we give a full blowup analysis of this limit, showing
that the two spheres are connected by a catenoidal neck. Besides its geometric
interest, the problem was studied as a simplified model in the theory of cell
membranes, see e.g. Berndl, Lipowsky, Seifert (Physical Review A, 1991).
