A graph Gamma is said to be locally primitive if, for each vertex alpha, th
e stabilizer in Aut Gamma of alpha induces a primitive permutation group on
the set of vertices adjacent to alpha. In 1978, Richard Weiss conjectured
that for a finite vertex-transitive locally primitive graph Gamma, the numb
er of automorphisms fixing a given vertex is bounded above by some function
of the valency of Gamma. In this paper we prove that the conjecture is tru
e for finite non-bipartite graphs provided that it is true in the case in w
hich Aut Gamma contains a locally primitive subgroup that is almost simple.