The evaporation of drops of pure water on soluble substrates of salt leads to the growth of unprecedented hollow peripheral deposits. We report here on the experimental determination and numerical simulation of the flows inside a drop in this configuration. The velocity field is measured experimentally by a microparticle image velocimetry technique and is simulated using the hydrodynamics equations solved by the finite-elements method. The flow is characterized by an inward motion at the beginning of the evaporation, that is progressively replaced by an outward motion until the end of the evaporation. The transition between the two flow regions takes the form of a stagnation line, where the radial component of the velocity vanishes. This line migrates from the periphery toward the center of the drop, inducing the progressive inversion of the flow. The comparison between the experiments and simulations leads to the following interpretation of this flow reversal. The dissolution of the substrate induces the creation of a surface tension gradient at the drop free-surface, leading to an outward Marangoni flow along the drop surface and an inward flow near the substrate. As the diffusion progressively makes the concentration more uniform, this surface-driven convection stops and is replaced by the standard coffee-stain outward capillary flow.