A mathematically exact method is presented for sampling conformations of polymer molecules in an external field with fixed energy or energy range in accord with the formulation of statistical mechanics for a microcanonical ensemble. As a consequence, conformations of negligible Boltzmann weight can be selectively eliminated from simulations for efficient calculation of macroscopic polymer properties. The method is applicable for conformations that are described by a stochastic differential equation along the contour length in the field-free situation. It is based on the concept of a stochastic bridge process for which a new stochastic differential equation is derived that has stipulations at both ends of the process. This idea is exploited on a pair of stochastic differential equations in the conformation vector X and an augmented variable Z which represents the running Boltzmann weight in the given field, transforming to a new pair of equations for which the terminal Boltzmann weight can be arbitrarily stipulated. The stochastic equation for the bridge involves solving the Fokker-Planck equation for the original stochastic pair. We demonstrate the method on the conformation of a ''Brownian'' polymer in a quadratic external field of varying strength. The stochastic differential equations for the bridge process in this case can be derived analytically. Sample conformations are displayed that satisfy exactly energy constraints either at fixed values or within a stipulated range. It is shown that polymer properties can be computed more efficiently and accurately with the bridge process simulations than by unconstrained process simulations. The bridge process approach presented here must be distinguished from other approaches such as umbrella sampling methods because of the former's ability to sample conformations exactly with stipulated energy constraints. (C) 1997 American Institute of Physics.