We present an approach to domain adaptation, addressing the case where data from the source domain is abundant, labelled data from the target domain is limited or non-existent, and a small amount of paired source-target data is available. The method is designed for developing deep learning models that detect the presence of medical conditions based on data from consumer-grade portable biosensors. It addresses some of the key problems in this area, namely, the difficulty of acquiring large quantities of clinically labelled data from the biosensor, and the noise and ambiguity that can affect the clinical labels. The idea is to pre-train an expressive model on a large dataset of labelled recordings from a sensor modality for which data is abundant, and then to adapt the model's lower layers so that its predictions on the target modality are similar to the original model's on paired examples from the source modality. We show that the pre-trained model's predictions provide a substantially better learning signal than the clinician-provided labels, and that this teacher-student technique significantly outperforms both a naive application of supervised deep learning and a label-supervised version of domain adaptation on a synthetic dataset and in a real-world case study on sleep apnea. By reducing the volume of data required and obviating the need for labels, our approach should reduce the cost associated with developing high-performance deep learning models for biosensors.