Distantly-labeled data can be used to scale up training of statistical models, but it is typically noisy and that noise can vary with the distant labeling technique. In this work, we propose a two-stage procedure for handling this type of data: denoise it with a learned model, then train our final model on clean and denoised distant data with standard supervised training. Our denoising approach consists of two parts. First, a filtering function discards examples from the distantly labeled data that are wholly unusable. Second, a relabeling function repairs noisy labels for the retained examples. Each of these components is a model trained on synthetically-noised examples generated from a small manually-labeled set. We investigate this approach on the ultra-fine entity typing task of Choi et al. (2018). Our baseline model is an extension of their model with pre-trained ELMo representations, which already achieves state-of-the-art performance. Adding distant data that has been denoised with our learned models gives further performance gains over this base model, outperforming models trained on raw distant data or heuristically-denoised distant data.