We recently presented the construction of a proportional counter for the absolute measurement of low-energy x-ray emission rates. Its main features were presented in a previous study, along with the testing of its performance as an absolute technique. Full details on its design, construction, and characterization processes are given in this paper. First, the analytical calculations and Monte Carlo simulations that led to the choice of optimal dimensions will be presented, followed by the study of the charge creation and the modeling of the electric field that confirmed that the criteria for a reproducible charge multiplication are met. Afterward, details on the electronics and gas control system will be provided. The dimensional measurements for the accurate determination of the solid angle will be described in detail. Special emphasis will be placed on the production of the detector window, which was made of reduced graphene oxide instead of beryllium. Some tests on the operation of the counter will be presented, namely, the measurement of spectra of several radionuclides that emit x-rays between 3 keV and 9 keV, as well as the fluorescence spectrum of chlorine, at 2.6 keV. A brief explanation of the determination of emission rates will be given, including the approach adopted for the treatment of the escape-peak area. The emission rate measured with the proportional counter was converted into activity concentration using the emission probabilities from the bibliography, in order to compare the results with those obtained by liquid scintillation counting, and both results were found to be in good agreement.