We have obtained échelle spectroscopy of 14 Population II objects selected from those previously observed by Bonifacio & Molaro (1997). For one object, HD 140283, we obtained exquisite data with the High Dispersion Spectrograph on the Subaru Telescope, with S/N exceeding 1000 per 0.018 Å pixel. Li abundances have been determined by spectral synthesis from both the 6708 Å resonance line and also from 6104 Å subordinate feature. Firm detections of the weak line have been made in seven objects, and upper limits are reported for the remainder. Our 6708 Å abundances agree with those reported by Bonifacio & Molaro with a mean difference of only 0.019 +/- 0.009 dex. Abundances from the 6104 Å line hint at a higher Li abundance than that determined from the resonance feature, but this evidence is mixed; the weakness of the 6104 Å line and the large number of upper limits make it difficult to draw firm conclusions. NLTE-corrections increase (rather than eliminate) the size of the (potential) discrepancy, and binarity appears unlikely to affect any abundance difference. The effect of multi-dimensional atmospheres on the line abundances was also considered, although it appears that use of 3-D (LTE) models could again act to increase the discrepancy, if one is indeed present.