In healthcare, it is common for the initial goal of modeling to be the prediction of critical but rare tasks (e.g. septic shock, cardiac arrest). The reality upon deployment of such a model is often different - the goal now is to assess the risk of a patient towards these (and other) critical events. The labels and their distribution of this auxiliary goal are different from the initial labels used for training the model. Continual Learning frameworks serve as an excellent way to update a model given new data, after it has been deployed in a production environment. We introduce CLARE, a Continual Learning framework which first pre-trains on a rare task (e.g. cardiac arrest), then updates according to the labels of assessed risk, collected from the clinicians in real time - a related task. We develop a novel replay-based method to sequentially learn from new data with a different label distribution. We compare our method to a model trained in a cumulative fashion as well as one that randomly replays earlier samples it has seen. We benchmark classification architectures on a simulated dataset as well as on a clinical dataset of physiological signals.