New, high-precision observations of the Li I lambda 6707 blend in the spectra of five halo stars with metallicities in the range -2.0 less than or equal to [Fe/H] less than or equal to -1.0 yield no positive detections of the lighter isotope at upper limits ranging from (6)Li/(7)Li less than or equal to 0.02 to (6)Li/(7)Li less than or equal to 0.13. In determining these limits, we removed two constraints that were previously imposed during comparison of the observed and synthetic line profiles. A reanalysis of our previous similar observations of six other halo stars with -2.9 less than or equal to [Fe/H] less than or equal to -1.1 was also carried out with this same procedure. The reanalysis confirms our earlier detection of (6)Li in HD 84937 with an isotopic ratio (6)Li/(7)Li = 0.08 +/- 0.04 (1 sigma), in agreement with Smith, Lambert, & Nissen, but converts our reported detection of (6)Li in HD 201891 to a comparable upper limit only, (6)Li/(7)Li less than or equal to 0.055. Among the 12 halo stars examined for (6)Li at high sensitivity by one or more of three different groups, positive results have been obtained only for the hottest dwarf, HD 84937, and possibly for the hottest subgiant, HD 160617, for which Nissen found (6)Li/(7)Li = 0.0173 +/- 0.012 (1 sigma). However, the less sensitive upper limits available for three slightly cooler dwarfs exceed the isotopic ratio at which 6Li is seen in HD 84937. In combination with Be and B abundances measured for some of these 12 stars, the very sparse (6)Li data suggest that a process such alpha + alpha reactions at low energies produced much of the (6)Li that was present at [Fe/H] approximate to -2.2, and that at least one of the following three assumptions is in error. (1) the (6)Li/Be ratio measured in HD 84937 is typical of halo gas at [Fe/H] = -2.2; (2) the Galactic (6)Li/Be ratio remained approximately invariant at -2.2 less than or similar to [Fe/H] less than or similar to -1.0; and (3) most of the original (6)Li in the probable subgiants HD 134169 and HD 201891 (with [Fe/H] = -1.0 and -1.1, respectively, and T(e) greater than or equal to 5800 K) has survived to the present day, as the observations of Pilachowski, Sneden, & Booth and the calculations of Chaboyer together suggest. Alternatively, the low (6)Li fractions required by the upper limits measured in HD 134169 and HD 201891 can be understood if both stars are dwarfs, not subgiants.