  Minimal algebraic surfaces of general type with the smallest possible
invariants have geometric genus zero and K^2=1 and are usually called
"numerical Godeaux surfaces". Although they have been studied by several
authors, their complete classification is not known. In this paper we classify
numerical Godeaux surfaces with an involution, i.e. with an automorphism of
order 2. We prove that they are birationally equivalent either to double covers
of Enriques surfaces, or to double planes of two different types: the branch
curve either has degree 10 and suitable singularities, originally suggested by
Campedelli, or is the union of two lines and a curve of degree 12 with certain
singularities. The latter type of double planes are degenerations of examples
described by Du Val and their existence was previously unknown; we show some
examples of this new type, computing also their torsion group.
