Richardson splitting applied to a consistent system of linear equations Cx = b with a singular matrix C yields to an iterative method x k+1 = Ax k + b where A has the eigenvalue one. It is known that each sequence of iterates is convergent to a vector x* = x* (x0) if and only if A is semi-convergent. In order to enclose such vectors we consider the corresponding interval iteration $$[x]^{k+1} = [A][x]^k+[b]$$ with ρ(|[A]|) = 1 where |[A]| denotes the absolute value of the interval matrix [A]. If |[A]| is irreducible we derive a necessary and sufficient criterion for the existence of a limit $$[x]^* = [x]^*([x]^0)$$ of each sequence of interval iterates. We describe the shape of $$[x]^*$$ and give a connection between the convergence of ( $$[x]^k$$ ) and the convergence of the powers $$[A]^k$$ of [A].