The challenge of controlled sampling of the conformations of internal sections of chain molecules, subject to constrained interatomic bond lengths and angles, is central to many areas of macromolecular science. A new method for overcoming this challenge via an internal configuration bias (ICB) Monte Carlo algorithm is described. It is demonstrated that the algorithm obeys the detail balance (microscopic reversibility) criterion necessary for performing rigorous molecular simulations in equilibrium ensembles. The algorithm is applied to a study of the molecular conformations of cyclic alkane molecules in a vacuum, where it is shown to be up to similar to 2 orders of magnitude more efficient than standard molecular dynamics simulation techniques. Qualitative transitions between constrained ring and flexible chain behavior are observed between 16 and 30 backbone atoms for local structure (torsion angle distribution) and between 30 and 50 backbone atoms for global ring dimensions.