We characterize the linear operators which preserve the factor rank of integer matrices. That is, if $mathcal{M}$ is the set of all $m{ imes}n$ matrices with entries in the integers and min($m,n$) > 1, then a linear operator T on $mathcal{M}$ preserves the factor rank of all matrices in $mathcal{M}$ if and only if T has the form either T(X) = UXV for all $X{in}mathcal{M}$, or $m=n$ and T(X)=$UX^tV$ for all $X{in}mathcal{M}$, where U and V are suitable nonsingular integer matrices. Other characterizations of factor rank-preservers of integer matrices are also given.