Recent experimental investigations of the kinetics of vesicle adsorption in solution on SiO2 demonstrate a thermally activated transition from adsorbed intact vesicles to a supported lipid bilayer. Our Monte Carlo simulations clarify the mechanism of this process. The model employed is an extension of the model used earlier to describe vesicle adsorption at room temperature. Specifically, it includes limitations of the adsorption rate by vesicle diffusion in the solution, and adsorption- and lipid-membrane-induced rupture of arriving and already adsorbed vesicles. Vesicles and lipid molecules, formed after rupture of vesicles, are considered immobile. With these ingredients, the model is able to quantitatively reproduce the temperature-dependent adsorption kinetics, including a higher critical surface concentration of intact vesicles for lower temperatures, and the apparent activation energy for the vesicle-to-bilayer transition E(a) approximately 5 kcal/mol.