The equilibrium conformation of a polymer molecule in an external field is often used in field theories to calculate macroscopic polymer properties of melts and solutions. We use a mathematical method called a Brownian bridge to exactly sample continuous polymer chains to end in a given state. We show that one can systematically develop such processes to sample specific polymer topologies, to confine polymers in a given geometry for its entire path, to efficiently generate high-probability conformations by excluding small Boltzmann weights, or to simulate rare events in a rugged energy landscape. This formalism can improve the polymer sampling efficiency significantly compared to traditional methods (e.g., Monte Carlo or Rosenbluth).