The technique of exact enumeration coupled with series analysis has been used to study the behaviour of the properties of long self-avoiding walks on a square lattice slab as the thickness (D) of the slab is varied. Scaling arguments due to Daoud and de Gennes (1977) predict the variation of mean-square end-to-end distance and of free energy with D. The results obtained are consistent with these scaling predictions for the mean-equare end-to-end distance, but suggest that the free-energy crossover exponent is closer to unity than the value (⁴/₃) predicted by scaling.