In this paper and in a subsequent one, we present relative and absolute measurements of photodetachment cross sections of negative gold ions in an RF trap. Several laser devices needed to be used. Absolute measurements are possible only when saturation occurs for a reasonable focusing of the laser and are deduced from the variations of photoelectron yield versus laser pulse energy. We describe the principle behind our measurements and we report on the results obtained in the energy range from threshold up to about above threshold. In this region, we essentially observe a slow departure from Wigner's law (always less than 50%).