Echelle spectra of six halo stars with -2.9 less-than-or-equal-to [Fe/H] less-than-or-equal-to - 1.4 and T(e) greater than or similar to 5650 K were recorded in the region of the Li I lambda6707 line, at S/N ratios 230 less-than-or-equal-to S/N < 500 and a resolving power of 120,000. Similar spectra were also obtained at the Ca I lambda6162 line, in order to determine the various contributions to the stellar line widths. Theoretical Li I line profiles were then calculated for several Li-6/Li-7 ratios and were compared to the observed profiles. The lighter isotope is detected in HD 201891 and HD 84937, at abundance ratios Li-6/Li-7 = 0.05 +/- 0.02 and 0.07 +/- 0.03, respectively. Upper limits on Li-6/Li-7 ranging from less-than-or-equal-to 0.03 to less-than-or-equal-to 0.10 are found for BD +3-degrees 740, BD +26-degrees 3578, HD 19445, and HD 140283. Our results for HD 84937 and HD 19445 agree with those of Smith, Lambert, & Nissen (1993). The observed Li-6 fractions are discussed in terms of stellar evolutionary models and the primordial lithium abundance.