  A detailed description of the structure of two-ended arc-transitive digraphs
is given. It is also shown that several sets of conditions, involving such
concepts as Property Z, local quasi-primitivity and prime out-valency, imply
that an arc-transitive digraph must be highly-arc-transitive. These are then
applied to give a complete classification of two-ended highly-arc-transitive
digraphs with prime in- and out-valencies.
