The adsorption of the strong polyelectrolyte sodium poly(styrenesulfonate) (NaPSS) to the air surface of water was investigated as a function of polymer concentration from the dilute regime to the beginning of the semidilute regime. Detailed segment profiles of the deuterated polymer were determined by neutron reflection (NR). Data were obtained for 0.67 and 2.50 M KCl. For two samples differing widely in molecular weight (1150 and 56.1 kg(mol), we find that with increasing polymer concentration the adsorbed amount first increases, reaches a maximum, and then decreases strongly. The PSS concentration at which the maximum is reached is dependent on both the molecular weight and the salt concentration in a manner that correlates with the chain overlap concentration. Regarding the segment profiles, at low polymer concentration the profiles are composed of a thin layer of high concentration at the air surface (trains), followed by a distinct second layer of much lower segment concentration that extends to larger depths into the liquid (loops and tails). Complementary X-ray reflection MR) revealed a localization of ions about 101 below the surface for dilute PSS concentration. This layer becomes more diffuse at higher PSS concentration, in conjunction with the decrease in PSS adsorbed amount measured by NR. This surprising behavior of the adsorbed amount with polymer concentration is not explained by current SCF theory treating the adsorption of strong polyelectrolytes to neutral surfaces. We discuss a few possible explanations for this desorption transition. Finally, we observe that the surface tension decreases monotonically with increasing concentration of PSS in bulk solution but is not correlated with the adsorbed amount of PSS at the surface.