  This paper proposes a simple strategy to simulate stochastic differential
equations (SDE) arising in constant temperature molecular dynamics. The main
idea is to patch an explicit integrator with Metropolis accept or reject steps.
The resulting `Metropolized integrator' preserves the SDE's equilibrium
distribution and is pathwise accurate on finite time intervals. As a corollary
the integrator can be used to estimate finite-time dynamical properties along
an infinitely long solution. The paper explains how to implement the patch
(even in the presence of multiple-time-stepsizes and holonomic constraints),
how it scales with system size, and how much overhead it requires. We test the
integrator on a Lennard-Jones cluster of particles and `dumbbells' at constant
temperature.
