This article describes the design and implementation of a variable timestep method for simulating time-reversible constrained dynamical systems. Based on the Adaptive Verlet method of Huang and Leimkuhler, and the SHAKE and RATTLE discretizations, the new method (VRATTLE) defines a mapping of the constraint manifold which preserves the reversible structure. It achieves this through the solution of a single additional scalar nonlinear equation at each timestep, together with the equations of constraint. As a nontrivial application, we simulate the dynamics of an elastic (inextensible, unshearable) rod undergoing large deformations and collisions with the sides of a bounding box. Numerical experiments indicate that adapting the stepsize using VRATTLE can smooth the numerical energy and improve the overall efficiency of the simulation.