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We discuss the interstellar absorption from many atomic and molecular species seen in high-resolution $HST$/STIS UV and high-S/N optical spectra of the moderately reddened B3-5~V star HD~62542. This remarkable sight line exhibits both very steep far-UV extinction and a high fraction of hydrogen in molecular form -- with strong absorption from CH, C$_2$, CN, and CO, but weak absorption from CH$^+$ and most of the commonly observed diffuse interstellar bands. Most of the material resides in a single narrow velocity component -- offering a rare opportunity to probe the primarily molecular core of a single interstellar cloud with little associated diffuse atomic gas. Detailed analyses of the spectra indicate that: (1) the molecular fraction in the main cloud is high [$f$(H$_2$) $>$ 0.8]; (2) the gas is fairly cold ($T_{\rm k}$ = 40--43 K, from the rotational excitation of H$_2$ and C$_2$); (3) the local hydrogen density $n_{\rm H}$ $\sim$ 1500 cm$^{-3}$ (from the C$_2$ excitation, the fine-structure excitation of C$^0$, and simple chemical models); (4) the unusually high excitation temperatures for $^{12}$CO and $^{13}$CO may be largely due to radiative excitation; (5) $N$(C$^+$):$N$(CO):$N$(C) $\sim$ 100:10:1; (6) the depletions of many elements are more severe than those seen in any other sight line, and the detailed pattern of depletions differs from those derived from larger samples of Galactic sight lines; and (7) the various neutral/first ion ratios do not yield consistent estimates for the electron density, even when the effects of grain-assisted recombination and low-temperaure dielectronic recombination are considered.