Glacial moulins (cylindrical meltwater drainage shafts) provide valuable insights into glacier dynamics, but are inaccessible and hazardous environments for humans to study. Exploring them using passive sensor probes has revealed their complex geometry, which has limited further exploration with passive sensor probes. To overcome these challenges, we propose a tethered robot capable of autonomously exploring and capturing data in glacial moulins. Our novel robot is equipped with a tether to support its motion. Combined with novel estimation and control algorithms, the tethered robot is able to safely and efficiently maneuver in confined, chimney-like structures, such as moulins. Laboratory and field experiments confirm the feasibility of the proposed design, showing successful localization in environments with no access to positional measurements. Field trials on the Mer de Glace glacier demonstrate the robot's capabilities, descending into the largest moulin to depths of 25 m, capturing valuable scientific data. Onboard sensors are used to capture data to reconstruct the moulin's three-dimensional geometry and two sampling mechanisms are presented and evaluated to extract samples from the icy surface of the moulin. Our results show promising potential for future exploration of moulins, demonstrating the effectiveness of our tethered robot for safely gathering data from these hazardous environments.