  We investigate the nature of the ground ring of c=1 string theory at the
special A-D-E points in the c=1 moduli space associated to discrete subgroups
of SU(2). The chiral ground rings at these points are shown to define the A-D-E
series of singular varieties introduced by Klein. The non-chiral ground rings
relevant to closed-string theory are 3 real dimensional singular varieties
obtained as U(1) quotients of the Kleinian varieties. The unbroken symmetries
of the theory at these points are the volume-preserving diffeomorphisms of
these varieties. The theory of Kleinian singularities has a close relation to
that of complex hyperKahler surfaces, or gravitational instantons. We speculate
on the relevance of these instantons and of self-dual gravity in c=1 string
theory.
