In support of our numerical simulations of self-diffusion D of chain molecules in the melt, we have used the pulsed-gradient spin-echo NMR method to make detailed D measurements in 15 liquid n-paraffins (8-60 carbons; molecular weight M=114-844) at eight temperatures T from 30 to 170 degrees C. We find that D-M-beta, with beta changing approximately linearly from -2.72 to -1.85 as T increases. Thus the apparent activation energies also rises linearly with log M. In the absence of molecular entanglements, Rouse kinetics predicts beta= -1, but Cohen-Turnbull-Bueche free-volume effects due to molecular chain ends add a further nonpower-law term, enhancing D increasingly at low M. The combined D vs M theory does, however, closely mimic a power law with the exponents observed at all our measurement temperatures. To obtain this result it is necessary to include in the free-volume term the dependence of the liquid's density on M and T, analytically modeled based on the incomplete literature data. The fitted density model itself is useful as a guide for the molecular modeling effort. The success of this approach makes it possible to deduce a small intrinsic thermal activation energy contained within the larger, free-volume dominated, apparent values.