The aim of the first protoype is to perform the measurement of energy and time resolution achievable in a liquid xenon PET scanner. It consists of an aluminum box filled with LXe, with two arrays of SiPMs on opposite sides, which will read out the scintillation light produced by a Na-22 calibration source placed in the middle. While some components of the prototype have been purchased as commercial solutions, for instance, the cryocooler that liquefies xenon or the gas filter, most of them have been designed specifically for this application. They include the gas system for the recirculation and purification of the gas, the vacuum vessel for thermal isolation, and the thermal links that connect the cryocooler with the xenon container and keep the temperature gradients under control. Especially crucial are the feedthroughs, which must bear high thermal stress and provide tightness to vacuum and liquid xenon pressure. A dedicated DAQ system has been also developed, which reads the output of the ASICs that digitize the signal and distribute and synchronize signals from different ASICs. The prototype has been assembled and has been taking data for the last year. During this time, on the one hand, we have tested the system, proving its excellent stability in the different phases of operation (filling, recirculation, data taking, recovery), we have characterised the full electronic chain and evaluated different kinds of SiPMs. On the other hand, we have measured the best energy resolution achieved so far in liquid xenon, using only scintillation light, that is 4.2% FWHM for 511-keV gammas. This result improves dramatically on the current knowledge of the field and suggests that no intrinsic fluctuations are present in the scintillation production in liquid xenon, contrary to what has been assumed so far. This result opens the possibility of building a PET with unprecedented energy resolution, with a better performance than the current state-of-the-art, especially for full-body PET applications.