  We prove existence and regularity of minimisers for the Canham-Helfrich
energy in the class of weak (possibly branched and bubbled) immersions of the
$2$-sphere. This solves (the spherical case) of the minimisation problem
proposed by Helfrich in 1973, modelling lipid bilayer membranes. On the way to
prove the main results we establish the lower semicontinuity of the
Canham-Helfrich energy under weak convergence of (possibly branched and
bubbled) weak immersions.
