Manufacturing dissimilar joints of aluminum and copper is a challenging task. However, friction stir welding (FSW) was found to be a suitable technique to produce aluminum–copper joints. Due to different electrical conductivities between aluminum and copper, an adjustment of the cross-section is required to realize electrical conductors free of resistive losses. Taking this into account, this paper presents initial results on the mechanical and electrical properties of friction stir butt welded aluminum and copper blanks having thicknesses of 4.7 mm and 3 mm, respectively. Three different approaches were investigated with the aim to produce sound welds with properties similar to those of the used base materials. Friction stir welding has been conducted at a welding speed of 450 mm/min. Subsequently, the welded specimens were subjected to metallographic analysis, tensile testing, and measurements of the electrical conductivity. The ultimate tensile force of the best joints was about 10 kN, which corresponds to joint efficiencies of approximately 72% of the aluminum base material. The analysis of electrical joint properties led to very promising results, so that the potential of FSW of Al–Cu butt joints with sheets having different thicknesses could be confirmed by the investigations carried out.