A fully-optimized Gaussian basis set is generated from a Monte Carlo simulated annealing procedure. The reliability of this stochastic procedure is assessed in generating of Gaussian basis sets for a set of atoms. After to confirm the reliability of the method, the basis set is expanded to include H, He, and the rest of the first- and second-rows atoms of the Periodic Table. Finally, a contraction scheme is chosen, and polarization and diffuse functions are added to the basis set. The proposed methodology has proved to be reliable and flexible to generate atom-optimized Gaussian basis sets with relatively low computational cost. The contracted basis set, augmented with polarization and diffuse functions, is used to calculate correlation energies of a training set of 106 molecular systems and it presents better performance than other basis sets of similar size. The proposed methodology can be used to generate Gaussian basis sets of increasing sizes.