The fabrication of dissimilar aluminum/copper joints for electrical application raises the challenges for conventional joining technologies. Within the solid-state processes, friction stir welding (FSW) provides numerous advantages to realize different joint configurations, especially by minimizing the heat input and hence the formation of brittle intermetallic phases. However, the joints also have to provide a high contact interface with firm bonding in order to provide a minimal contact resistance. Therefore, joints of 1 mm ENCW004A and EN AW1050A with a controlled melt layer formation were produced by friction stir spot welding (FSSW). By using a pinless tool and the positioning of copper as the upper joining partner, local melt formation at the interface with a eutectic composition was promoted without significant intermixing, resulting in wetting of the aluminum and a contact area increase. The rotational speed was varied between 1800–2400 rpm, in which range samples with up to 300-μm-thick melt layers were produced. The wetting effect at the interface shows a positive influence on the shear strength with ductile failure behavior even at high layer thickness. The microstructural composition at the interface showed a eutectic composition for small layer thickness and an inhomogeneous composition with hypo- and hypereutectic solidification structures for higher thickness values. However, the formation of intermetallic compounds other than CuAl2 was mostly inhibited by the short process times and high cooling rate.