The asymptotic behavior of mean velocity and integral parameters in flat plate turbulent boundary layers under zero pressure gradient are studied for Reynolds numbers approaching infinity. Using the classical two-layer approach of Millikan, Rotta, and Clauser with a logarithmic velocity profile in the overlap region between "inner" and "outer" layers, a fully self-consistent leading-order description of the mean velocity profile and all integral parameters is developed. It is shown that this description fits most high Reynolds number data, and in particular their Reynolds number dependence, exceedingly well; i.e., within experimental errors. (c) 2007 American Institute of Physics.