Improved neutron capture cross section data for transuranic and minor actinides are essential for assessing possibilities for next generation reactors and advanced fuel cycles. The Measurement of Actinide TRAnsmutation (MANTRA) project aims to make a comprehensive set of energy integrated neutron capture cross section measurements for all relevant isotopes from Th to Cf. The ability to extract these cross sections relies on the use of Accelerator Mass Spectrometry (AMS) to analyze isotopic concentrations in samples irradiated in the Advanced Test Reactor (ATR).The AMS measurements were performed at the Argonne Tandem Linear Accelerator System (ATLAS) and required a number of key technical developments to the ion source, accelerator, and detector setup. In particular, a laser ablation material injection system was developed at the electron cyclotron resonance ion source. This system provides a more effective method to produce ion beams from samples containing only 1% actinide material and offers some benefits for reducing cross talk in the source.A series of four actinide measurements are described in this dissertation. These measurements represent the most substantial AMS work attempted at ATLAS and the first results of the MANTRA project. Isotopic ratios for one and two neutron captures were measured in each sample with total uncertainties around 10%. These results can be combined with a MCNP model for the neutron fluence to infer actinide neutron capture cross sections.