We report our observations of the λ6708A absorption lines of interstellar
6LiI and
7LiI in the direction of ζ Oph, conducted at the ESO 3.6m Telescope linked via fibre optics to the Echelle Coude Spectrometer. Our LiI data revealed after a careful reduction to be of a higher quality than those previously obtained on the same line of sight, with a resolving power λ/{DELTA}λ=100000 and a signal-to-noise ratio S/N=7500 per pixel. We detect one main A and one weak B absorbing interstellar components in both LiI and KI spectra. Using sophisticated profile fitting techniques, we are able to derive two isotopic ratios, (
7Li/
6Li)
A=8.6±0.8(±1.4 systematics), and (
7Li/
6Li)
B=1.4
+1.2–0.5(±0.6 systematics). We show that the results of Meyer et al. (
1993ApJ...409L..61M) for the
7Li/
6Li ratio toward ζ Oph,
7Li/
6Li=6.8
+1.4–1.7, and toward ζ Per for similar reasons
7Li/
6Li=5.5
+1.3–1.1, are biased because only one interstellar component was taken into account in their analysis whereas two at least are clearly present. As well, we find that an average value of the
7Li/
6Li ratio on the line of sight to ζ Oph should lie closer to 8-9 than to 6-7. The ratio derived by Ferlet & Dennefeld (
1984A&A...138..303F) toward ζ Oph is unexplained,
7Li/
6Li>25, eventhough
6Li was not detected for that time. The (
7Li/
6Li)
A ratio derived here is in slight agreement with the
7Li/
6Li ratio measured toward ρ Oph,
7Li/
6Li=11.1±2, and strongly argues for the existence of an extra source of
7Li in the Galaxy. The (
7Li/
6Li)
B ratio is not readily explained either by an instrumental contamination, by the presence of additional clouds on the line of sight, or even by a massive interaction of cosmic rays with the material of cloud B. Although none of these hypotheses seems entirely satisfactory, each one of them implies that the value derived in cloud A should be higher than what is observed, i.e. (
7Li/
6Li)
A>8.6.