Diffuse interstellar bands (DIBs) have been detected for the first time along the peculiar translucent line of sight toward HD 62542, which passes through a diffuse cloud core. Although only a small fraction (18 out of more than 300) of generally weak DIB features have been shown to correlate with C₂ and C₃ (the "C₂ DIBs"), it is predominantly these DIBs that are observed toward HD 62542. The typically strong DIBs λλ5780 and 5797 are detected but are significantly weaker than toward other lines of sight with similar reddening. Other commonly observed DIBs (such as λλ4430, 6270, and 6284) remain noticeably absent. These observations further support the suggestion that the line of sight toward HD 62542 crosses only the core of a diffuse cloud and show that the correlation between the C₂ DIBs and small carbon chains is maintained in environments with very large fractions of molecular hydrogen, f > 0.8. A comparison of CH, CN, C₂, and C₃ column densities and C₂ DIB strengths toward HD 62542, HD 204827, and HD 172028 suggests that the line of sight toward HD 204827 passes through a diffuse cloud core similar to that seen toward HD 62542, as well as what might be referred to as a diffuse cloud envelope. This indicates that the bare core toward HD 62542 may not have significantly different relative chemical abundances from other diffuse cloud cores and that the C₂ DIBs may serve as a diagnostic of such cores.