  We prove convergence of Goodwillie--Weiss' embedding calculus for spaces of
diffeomorphisms of surfaces and for spaces of arcs in surfaces. This exhibits
the first nontrivial class of examples for which embedding calculus converges
in handle codimension less than three and results in particular in a
description of the mapping class group of a closed surface in terms of derived
maps of modules over the framed $E_2$-operad.
