The structure of the Langmuir-Gibbs films of normal alkanes C-n of length n = 12-21 formed at the surface of aqueous solutions of CTAB surfactants, m = 14, 16, and 18, was studied by surface-specific synchrotron X-ray methods. At high temperatures, a laterally disordered monolayer of mixed alkane molecules and surface-adsorbed surfactant tails is found, having thicknesses well below those of the alkanes' and surfactant tails' extended length. The mixed monolayer undergoes a freezing transition at a temperature T-s(n,m), which forms, for n <= m + 1, a crystalline monolayer of mixed alkane molecules and surfactant tails. For n >= m + 2, a bilayer forms, consisting of an upper pure-alkane, crystalline monolayer and a lower liquidlike monolayer. The crystalline monolayer in both cases consists of hexagonally packed extended, surface-normal-aligned chains. The hexagonal lattice constant is found to decrease with increasing n. The films' structure is discussed in conjunction with their thermodynamic properties presented in an accompanying paper.