We present a deep active contour model for detecting and delineating glacier calving fronts from satellite imagery. Contrary to existing deep learning-based calving front detectors, our model does not perform an intermediate segmentation or pixel-wise edge detection, but instead directly predicts the contour parametrized by a fixed number of vertices. The model works by first deriving feature maps from an input image, and then updating an initial contour in an iterative fashion. Evaluating on the CALFIN dataset, which maps calving fronts in Greenland, our model outperforms existing approaches. Code for the experiments and animated predictions can be found at https://github.com/khdlr/deep-acm