  We present a novel integrable non-autonomous partial differential equation of
the Schwarzian type, i.e. invariant under M\"obius transformations, that is
related to the Korteweg-de Vries hierarchy. In fact, this PDE can be considered
as the generating equation for the entire hierarchy of Schwarzian KdV
equations. We present its Lax pair, establish its connection with the SKdV
hierarchy, its Miura relations to similar generating PDEs for the modified and
regular KdV hierarchies and its Lagrangian structure. Finally we demonstrate
that its similarity reductions lead to the {\it full} Painlev\'e VI equation,
i.e. with four arbitary parameters.
