We present observations of the LiI 6708Å doublet in the ζ Oph sightline, obtained at a resolution of ~106 and a signal-to-noise ratio of ~1200, together with supplementary observations of KIλ7699 and the NaIλ3302 doublet. These observations marginally resolve the main `-15kms-1' system into its two principal clouds; we model the data using standard physical assumptions, but in a statistically rigorous manner, taking fully into account the hyperfine, doublet and isotopic structure for each species, in each cloud, using a new code, VAPID. The average 7Li/6Li ratio determined in this sightline is 1.12+/-0.20dex (68 per cent confidence interval), in excellent agreement with the solar-system (meteoritic) value. The ratios in the individual clouds are determined with less precision, but are also consistent with the solar-system value, as is the total lithium abundance (with little evidence for depletion). The thermal and `turbulent' broadening in the two clouds is discussed on the basis of observed line widths.