  We consider the cycle structure of a random permutation $\sigma$ chosen
uniformly from the symmetric group, subject to the constraint that $\sigma$
does not contain cycles of length exceeding $r.$ We prove that under suitable
conditions the distribution of the cycle counts is approximately Poisson and
obtain an upper bound on the total variation distance between the distributions
using Stein's method of exchangeable pairs. Our results extend the recent work
of Betz, Sch\"{a}fer, and Zeindler.
