  Given a branched Willmore immersion from a closed Riemann surface, we show
that Bryant's quartic is holomorphic. Consequently, this quartic vanishes when
the underlying surface is a sphere and we obtain the full classification of
branched Willmore spheres. To do so, we show that the asymptotic expansion in
the $C^2$-topology of the conformal Gauss map at a branched point is a null
straight line.
