The Marangoni contraction of sessile drops of a binary mixture of a volatile and a nonvolatile liquid has been investigated experimentally and theoretically. The origin of the contraction is the locally inhomogeneous evaporation rate of sessile drops. This leads to surface tension gradients and thus to a Marangoni flow. Simulations show that the interplay of Marangoni flow, capillary flow, diffusive transport, and evaporative losses can establish a quasistationary drop profile with an apparent nonzero contact angle even if both liquid components individually wet the substrate completely. Experiments with different solvents, initial mass fractions, and gaseous environments reveal a previously unknown universal power-law relation between the apparent contact angle and the relative undersaturation of the ambient atmosphere: theta(app) similar to (RHeq - RH)(1/3). This experimentally observed power law is in quantitative agreement with simulation results. The exponent can also be inferred from a scaling analysis of the hydrodynamic-evaporative evolution equations of a binary mixture of liquids with different volatilities.