This paper describes the iterative design, training, validation, and statistical evaluation of a Convolutional Neural Network (CNN) for the estimation of an autonomous driving system’s (ADS) position and orientation (pose) using Light Detection and Ranging (LiDAR) data. Performance assessment is carried out in a structured, static lab environment using a LiDAR-equipped rover moving along a fixed, repeated trajectory. The LiDAR CNN performance is compared against a Landmark-Based Localization method (LBL). The LiDAR CNN and LBL have similar sample error standard deviations for position estimates. In our current implementation, the LiDAR CNN does not yet incorporate data from inertial measurement units and therefore shows higher heading angle estimation error.