The advent of Artificial Intelligence (AI) has im-pacted all aspects of human life. One of the concrete examples of AI impact is visible in radio positioning. In this article, for the first time we utilize the power of AI by training a Convolutional Neural Network (CNN) using 5G New Radio (NR) fingerprints consisting of beamformed Channel State Information (CSI). By observing CSI, it is possible to characterize the multipath channel between the transmitter and the receiver, and thus provide a good source of spatiotemporal data to find the position of a User Equipment (UE). We collect ray-tracing-based 5G NR CSI from an urban area. The CSI data of the signals from one Base Station (BS) is collected at the reference points with known positions to train a CNN. We evaluate our work by testing: a) the robustness of the trained network for estimating the positions for the new measurements on the same reference points and b) the accuracy of the CNN-based position estimation while the UE is on points other than the reference points. The results prove that our trained network for a specific urban environment can estimate the UE position with a minimum mean error of 0.98 m.