Observations of glacier melt and runoff are of fundamental interest in the study of glaciers and their interactions with their environment. Considerable recent interest has developed around distributed acoustic sensing (DAS), a sensing technique which utilizes Rayleigh backscatter in fiber optic cables to measure the seismo-acoustic wavefield in high spatial and temporal resolution. Here, we present data from a month-long, 9 km DAS deployment extending through the ablation and accumulation zones on Rhonegletscher, Switzerland, during the 2020 melt season. While testing several types of machine learning (ML) models, we establish a regression problem, using the DAS data as the dependent variable, to infer the glacier discharge observed at a proglacial stream gauge. We also compare two predictive models that only depend on meteorological station data. We find that the seismo-acoustic wavefield recorded by DAS can be utilized to infer proglacial discharge. Models using DAS data outperform the two models trained on meteorological data with mean absolute errors of 0.64, 2.25 and 2.72 m³ s^-1, respectively. This study demonstrates the ability of in situ glacier DAS to be used for quantifying proglacial discharge and points the way to a new approach to measuring glacier runoff.