To reconstruct magnetic resonance (MR) images from undersampled Cartesian k-space data, we propose an algorithm based on two deep-learning architectures: (1) a multi-layer perceptron (MLP) that estimates a target image from 1D inverse Fourier transform (IFT) of k-space; and (2) a convolutional neural network (CNN) that estimates the target image from the estimated image of the MLP. The MLP learns the relationship between 1D IFT of undersampled k-space which is transformed along the frequency-encoding direction and the target fully-sampled image. The MLP is trained line by line rather than by a whole image, because each frequency-encoding line of the 1D IFT of k-space is not correlated with each other. It can dramatically decrease the number of parameters to be learned because the number of input/output pixels decrease from N2 to N. The next CNN learns the relationship between an estimated image of the MLP and the target fully-sampled image to reduce remaining artifacts in the image domain. The proposed deep-learning algorithm (i.e., the combination of the MLP and the CNN) exhibited superior performance over a single MLP and a single CNN. And it outperformed the comparison algorithms including CS-MRI, DL-MRI, a CNN-based algorithm (denoted as Wang’s algorithm), PANO, and FDLCP in both qualitative and quantitative evaluation. Consequently, the proposed algorithm is applicable up to a sampling ratio of 25% in Cartesian k-space.