This paper deals with the problem of localization in a cellular network in a dense urban scenario. Global Navigation Satellite Systems typically perform poorly in urban environments, where the likelihood of line-of-sight conditions between the devices and the satellites is low, and thus alternative localization methods are required for good accuracy. We present a deep learning method for localization, based merely on pathloss, which does not require any increase in computation complexity at the user devices with respect to the device standard operations, unlike methods that rely on time of arrival or angle of arrival information. In a wireless network, user devices scan the base station beacon slots and identify the few strongest base station signals for handover and user-base station association purposes. In the proposed method, the user to be localized simply reports such received signal strengths to a central processing unit, which may be located in the cloud. For each base station we have good approximation of the pathloss at every location in a dense grid in the map. This approximation is provided by RadioUNet, a deep learning-based simulator of pathloss functions in urban environment, that we have previously proposed and published. Using the estimated pathloss radio maps of all base stations and the corresponding reported signal strengths, the proposed deep learning algorithm can extract a very accurate localization of the user. The proposed method, called LocUNet, enjoys high robustness to inaccuracies in the estimated radio maps. We demonstrate this by numerical experiments, which obtain state-of-the-art results.