Recent work has shown that diffusion models memorize and reproduce training data examples. At the same time, large copyright lawsuits and legislation such as GDPR have highlighted the need for erasing datapoints from diffusion models. However, retraining from scratch is often too expensive. This motivates the setting of data unlearning, i.e., the study of efficient techniques for unlearning specific datapoints from the training set. Existing concept unlearning techniques require an anchor prompt/class/distribution to guide unlearning, which is not available in the data unlearning setting. General-purpose machine unlearning techniques were found to be either unstable or failed to unlearn data. We therefore propose a family of new loss functions called Subtracted Importance Sampled Scores (SISS) that utilize importance sampling and are the first method to unlearn data with theoretical guarantees. SISS is constructed as a weighted combination between simpler objectives that are responsible for preserving model quality and unlearning the targeted datapoints. When evaluated on CelebA-HQ and MNIST, SISS achieved Pareto optimality along the quality and unlearning strength dimensions. On Stable Diffusion, SISS successfully mitigated memorization on nearly 90% of the prompts we tested. We release our code online.