Nucleation rates for the freezing of molten clusters of salt were determined in molecular dynamics simulations at 400, 500, 525, 550, and 580 K. These were analyzed in terms of the free energy of formation of the critical nucleus implied by the classical theory of homogeneous nucleation and by the diffuse interface theory of Granasy. Both the classical prefactor based on activated diffusion across the interface and the Grant-Gunton prefactor were examined. When each formulation of nucleation theory was adjusted to force J(T) to agree with the simulation at 525 K, the calculated rates corresponding to the four combinations of prefactor and exponential factor diverged rapidly from each other as the temperature departed from 525 K. This extreme incompatibility of the different formulations of nucleation theory was due as much to the different prefactors as to the different nucleation barriers. Such an incompatibility has been paid little heed, partly because it is less evident in studies at less extreme supercooling. Of the formulations considered, the classical nucleation theory with the classical prefactor and the diffuse interface theory with the Grant-Gunton prefactor were ruled out by the molecular dynamics simulations in combination with a criterion to estimate freezing rates at the evaporative cooling temperature. Nevertheless, this result, considered in light of the known flaws of the classical nucleation theory, suggests that the Grant-Gunton prefactor is excessively high. Also, the classical prefactor, which is known not to be universally applicable, appears to err in the opposite direction. Although theorists have devoted most of their efforts on nucleation in condensed phases to the free energy barrier, it is clear that refinement of the prefactor is of comparable importance.