A colorization method based on a fully convolutional neural network for grayscale images is presented in this paper. The proposed colorization method includes color space conversion, grayscale image preprocessing and implementation of improved U-Net network. The training and operating of the U-Net network take place for images represented in the space of the Lab color model. The trained U-Net network integrates realistic colors (generate data of a and b components) into grayscale images based on L-component data of the Lab color model. Median cut method of quantization is applied to L-component data before the training and operating of the U-Net network. Logistic activation function is applied to normalized results of convolution layers of the U-Net network. The proposed colorization method has been tested on ImageNet database. The evaluation results of the proposed method according to various parameters are presented. Colorization accuracy by the proposed method reachers more than 84.81%. The colorization method proposed in this paper is characterized by optimized architecture of convolution neural network that is able to train on a limited image set with a satisfactory training duration. The proposed colorization method can be used to improve the image quality and restoring data in the development of computer vision systems. The further research can be focused on the study of a technique of defining optimal number of the gray levels and the implementation of the combined quantization methods. Also, further research can be focused on the use of HSV, HLS and other color models for the training and operating of the neural network.