Living cells exhibit a complex mechanical behavior, whose underlying mechanisms are still largely unknown. Emerging from the molecular structure and dynamics of the cytoskeleton, the mechanical behavior comprises "passive" viscoelastic material properties and "active" contractile prestress. To directly investigate the connection between these quantities at the single-cell level, we here present the combination of atomic force microscopy (AFM) with traction force microscopy (TFM). With this combination, we simultaneously measure viscoelastic material parameters (stiffness, fluidity) and contractile prestress of adherent fibroblast and epithelial cells. Although stiffness, fluidity, and contractile prestress greatly vary within a cell population, they are highly correlated: stiffer cells have a lower fluidity and a larger prestress than softer cells. We show that viscoelastic material properties and contractile prestress are both governed by the activity of the actomyosin machinery. Our results underline the connection between a cell's viscoelastic material properties and its contractile prestress and their importance in cell mechanics.