Thin films of Indium-Tin-Oxide placed on high-quality glass substrates are on the one hand electrically conductive and on the other hand optically transparent. They have high homogeneity of sheet resistance, which allows using them as flat antennas for mobile communications and other applications. Knowledge of law of a current distribution along the antenna axis is of great importance for understanding of physical processes in antennas. This knowledge allows defining the main characteristics of the antennas. The flat transparent antenna is the linear antenna with nonzero boundary conditions. Similarly, in the case of a flat transparent antenna it is necessary first of all take into account the surface impedance. The experimental results confirmed the nature of the current distribution along the antenna and demonstrated the usefulness of a smooth transition from the wide transparent film to the central wire of the cable that to improve matching and to create a uniform current distribution across the whole width of the antenna.