The discovery of the second highly $r$-process-enhanced, extremely metal-poor star, CS 31082-001 ([Fe/H] $= -2.9$) has provided a powerful new tool for age determination, by virtue of the detection and measurement of the radioactive species uranium and thorium. One of the serious limitations of this approach, however, is that predictions of the production ratio of U and Th have not been made in the context of a realistic astrophysical model of the $r$-process. We have endeavored to produce such a model, based on the ``neutrino winds'' that are expected to arise from the nascent neutron star of a core-collapse supernova. The mass-integrated $r$-process yields, obtained by assuming a simple time evolution of the neutrino luminosity, are compared to the available spectroscopic elemental abundance data of CS 31082-001. As a result, the ``age'' of this star is determined to be $14.1 \pm 2.5$ Gyr, in excellent agreement with lower limits on the age of the universe estimated by other dating techniques, as well as with other stellar radioactive age estimates. Future measurements of Pt and Pb in this star, as well as expansion of searches for additional $r$-process-enhanced, metal-poor stars (especially those in which both U and Th are measurable), are of special importance to constrain the current astrophysical models for the $r$-process.

23 pages, 7 figures, accepted for publication in ApJ