Synthetic aperture radar (SAR) images often interfere with speckle artifacts that have a great impact on subsequent processing and analysis operations. To remove speckle artifacts, this article introduces a hybrid denoising approach by using a convolutional neural network (CNN) and consistent cycle spinning (CCS) in the nonsubsample shearlet transform (NSST) domain. First, we apply NSST to a noisy SAR image to gain low- and high-frequency coefficients. Second, we adopt a learned deep CNN model to eliminate the speckle noise in the low-frequency coefficients, which retains more contour information. Third, we employ CCS to enhance the high-frequency coefficients, which preserves more details of the original SAR image. Finally, we obtain the denoised image by using inverse NSST applied to the denoised coefficients. Compared with state-of-the-art algorithms, the results of the experiment indicate that our method not only achieves better speckle removal performance but also maintains more detailed information retention.