This paper introduces a deep learning network that reconstructs low-dose CT images into CT images of a high quality comparable to adaptive statistical iterative reconstruction (ASIR) as fast as filtered backprojection (FBP). Fully convolutional networks (FCNs) are adopted to denoise the low-dose CT images reconstructed with FBP. In contrast to patch-based convolutional neural networks (CNNs), we train the FCN-based denoising network with full-size images, which is computationally efficient due to the reuse of feature maps from the lower layers. To guarantee that the resultant high-quality images are consistent with the input images, a CNN-based classifier is added to the denoising network during the training phase. The classifier incorporates the CT noise model and evaluates the consistency between the images reconstructed with FBP and those of the denoising network. This supplementary structure makes the entire network a class of generative adversarial networks (GANs). For training and testing the network, we use a dataset of 18 patients who have undergone abdominal low-dose CT with both FBP and ASIR, which we split into a training set of 12 patients and a validation set of the remaining 6 patients. After being trained with FBP and ASIR image pairs, the GAN successfully recovers the high-quality images from the noisy CT images reconstructed with FBP. The network, by using a moderate GPU, is computationally efficient in recovering each image within 0.1 second. It is also remarkable that the GAN successfully preserves the image details, whereas ASIR is known for its occasional failure to recover small low-contrast features.