If a material grows on another material with a largely different lattice constant, which of the two adapts for an energetically favorable growth? To tackle this question, we investigate the growth of Ag on Cu(111) by variable temperature scanning tunneling microscopy. The structures grown between 120 and 170 K are remarkably different from those grown between 200 and 340 K. The low-temperature structure is rectangular-like and consists of stacked rods, 7 to 8 Ag atoms long, which form a superstructure without long-range order. This structure covers the whole surface prior to nucleation of further layers. The high-temperature structure is hexagonal and consists of misfit dislocations forming 8 × 8 to 10 × 10 superstructures. For this structure, second layer nucleation sets in far before the closure of the first monolayer. While both structures are driven by the large lattice misfit between the two materials, the growing Ag layer adapts to the Cu surface at low temperature, while the Cu surface adapts to the growing Ag layer at higher temperature.