We revisit the abundances of neutron-capture elements in the metal-poor ([Fe/H] = -2.9) r-process-rich halo star CS 31082-001. Partly motivated by the development of the new near-ultraviolet Cassegrain U-band Efficient Spectrograph for the Very Large Telescope, we compiled an expanded line list for heavy elements over the range 3000-4000 Å, including hyperfine structure for several elements. Combining archival near-ultraviolet spectra of CS 31082-001 from the Hubble Space Telescope and the Very Large Telescope, we investigate the abundances and nucleosynthesis of 35 heavy elements (Ge, Sr, Y, Zr, Nb, Mo, Ru, Rh, Pd, Ag, Cd, Sn, Ba, La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu, Hf, Os, Ir, Pt, Pb, Bi, Th, and U). Our analysis includes the first abundance estimates for tin, holmium, and ytterbium from these data, and the first for lutetium from ground-based data, enabling a more complete view of the abundance pattern of this important reference star. In general, the r-process-dominated elements are as enhanced as those in the Sun, particularly for elements with Z ≥ 56 (Ba and heavier). However, the abundances for the lighter elements in our sample, from Ge to Sn (31 ≤ Z ≤ 50), do not scale with the solar abundance pattern. Moreover, the Ge abundance is deficient relative to solar, indicating that it is dominantly an iron-peak rather than neutron-capture element. Our results (or upper limits) on Sn, Pt, Au, Pb, and Bi all pose further questions, prompting further study on the origin and evolution of the known r-rich and actinide-rich, metal-poor stars.