In this paper, we study the hitherto unexplored evolution of the size distribution of 185 functionally defined urban areas in Brazil between 1907 and 2008. We apply the traditional power law and the Gabaix-Ibragimov improved estimation method. We find that the power parameter of the size distribution of the 100 largest urban areas increases towards unity, that is, the rank-size rule. Seventeen thousand and twenty cointegration tests of pairwise non-stationary growth paths of cities point to evidence of a common stochastic trend that strengthens throughout the century. Thus, while Zipf's law and Gibrat's law did not formally hold in Brazil's past, these laws are becoming steadily more appropriate descriptions of Brazil's increasingly unequal size distribution of cities.