A new method for constructing probability tables in the Unresolved Resonance Region (URR) has been developed. This new methodology is an extensive modification of the Single-Level Breit-Wigner (SLBW) resonance-pair sequence method commonly used to generate probability tables in the URR. Using a Monte Carlo process, many resonance-pair sequences are generated by sampling the average resonance parameter data for the unresolved resonance region from the ENDF data file. The resonance parameters are then converted to the Reich-Moore format to take advantage of the more robust R-Matrix Limited (RML) format. For each sampled set of resonance-pair sequences, the temperature-dependent cross sections are calculated on a small grid around the energy of reference using the RML formalism and the Leal-Hwang Doppler broadening methodology. The effective cross sections calculated at the energy of reference are then used to construct probability tables in the unresolved resonance region. The RML cross section reconstruction algorithm has been rigorously tested for a variety of isotopes, including O-16, F-19, Cl-35, Fe-56, Cu-63, and Cu-65. The new URR method also produced normalized cross-section factor probability tables for U-238 that were found to be in agreement with current standards. The modified U-238 probability tables were shown to produce k-eff results in excellent agreement with several standard benchmarks, including the IEU-MET-FAST-007, IEU-MET-FAST-003, and IEU-COMP-FAST-004 benchmarks.