  An element $\phi$ of the outer automorphism group $\Out(\f)$ of the rank $n$
free group $F_n$ is {\it polynomially growing} if the word lengths of conjugacy
classes in $\f$ grow at most polynomially under iteration by $\phi$. It is {\it
unipotent} if additionally its action on the first homology of $\f$ with
integer coefficients is unipotent. In particular, if $\phi$ is polynomially
growing and acts trivially on first homology with coefficients the integers mod
3 then $\phi$ is unipotent and also every polynomially growing element has a
positive power that is unipotent. We solve the conjugacy problem in $\Out(\f)$
for the subset of unipotent elements. Specifically, there is an algorithm that
decides if two such are conjugate in $\Out(\f)$.
