A significant fraction of the experimental works on the rheology of polymer melts include an attempt to find the zero-shear-rate viscosity eta(0). This is done for good reasons, because eta(0) is a limiting property that depends only on thermodynamic variables and, importantly, the molecular and supermolecular structure of the melt. As with all limiting properties, eta(0) is impossible to measure directly. Fortunately with many melts, it can be estimated from viscosity measurements at very low shear rates or frequencies, but still remains one of those properties that becomes in the limit very prone to error. The common approach is to use a set of frequency- or shear-rate-dependent data and extrapolate to find eta(0). As with any extrapolation, the major question is the function used for the extrapolation. This question is addressed in some detail in this article. The question of which function to use was discarded in favor of using a large sample of 20 equations of many functional forms. This sample of randomly chosen equations was used to generate a set of eta(0) values, and the statistics of this distribution were examined, in the usual fashion, by description with an analytical probability density function that gives a high probability of being a likely generator of the data. In addition, a weighted average was proposed, where the weighting factor takes into account the quality of the fit. For testing these ideas, the room temperature melts of poly(vinyl isobutyl ether), poly(isobutylene), and poly(dimethyl siloxane) were used. The eta(0) of the latter was reachable; for the other resins, a falling ball technique was attempted.