We propose SEANet (Sound Enhancement Adversarial Network), an audio-to-audio generative model that learns to denoise and enhance old music recordings. Our model internally converts its input into time-frequency representation by means of a short-time Fourier transform (STFT), and processes the resulting spectrogram using a convolutional neural network. The network is trained with both reconstructive and adversarial objectives on a synthetic noisy music dataset, which is created by mixing clean music with real noise samples extracted from quiet segments of old recordings. We evaluate our method both quantitatively on held-out test examples of the synthetic dataset, and qualitatively by human rating on samples of actual historical recordings. Our results show that the proposed method is effective in removing noise, while preserving the musical quality and details of the original.