In this article we give a new proof of Ng\^o's Geometric Stabilisation
Theorem, which implies the Fundamental Lemma. This is a statement which relates
the cohomology of Hitchin fibres for a quasi-split reductive group scheme G
to the cohomology of Hitchin fibres for the endoscopy groups Hκ. Our
proof avoids the Decomposition and Support Theorem, instead the argument is
based on results for p-adic integration on coarse moduli spaces of
Deligne-Mumford stacks. Along the way we establish a description of the inertia
stack of the (anisotropic) moduli stack of G-Higgs bundles in terms of
endoscopic data, and extend duality for generic Hitchin fibres of Langlands
dual group schemes to the quasi-split case.Comment: 49 page