  We determine the absolute differential Galois group of the field
$\mathbb{C}(x)$ of rational functions: It is the free proalgebraic group on a
set of cardinality $|\mathbb{C}|$. This solves a longstanding open problem
posed by B.H. Matzat. For the proof we develop a new characterization of free
proalgebraic groups in terms of split embedding problems, and we use patching
techniques in order to solve a very general class of differential embedding
problems. Our result about $\mathbb{C}(x)$ also applies to rational function
fields over more general fields of coefficients.
