We present an efficient approach for reducing the statistical uncertaintyassociated with direct Monte Carlo simulations of the Boltzmann equation.As with previous variance-reduction approaches, the resulting relativestatistical uncertainty in hydrodynamic quantities (statistical uncertainty normalized by thecharacteristic value of quantity of interest) is smalland independent of the magnitude of the deviation from equilibrium,making the simulation of arbitrarily small deviations from equilibriumpossible. In contrast to previous variance-reduction methods, themethod presented here is able to substantially reduce variance withvery little modification to the standard DSMC algorithm. This is achievedby introducing an auxiliary equilibrium simulation which, via an importanceweight formulation, uses the same particle data as the non-equilibrium(DSMC) calculation; subtracting the equilibrium from the non-equilibriumhydrodynamic fields drastically reduces the statistical uncertaintyof the latter because the two fields are correlated. The resulting formulation is simple to code and provides considerable computational savings for a wide range of problems of practical interest. It is validated by comparing our results with DSMC solutions for steadyand unsteady, isothermal and non-isothermal problems; in all casesvery good agreement between the two methods is found.