  Results are presented for our search for warm-hot gas towards the quasar Q
0329-385. We identify ten OVI systems of which two are within 5000 km/s of z_em
and a third one should be of intrinsic origin. The seven remaining systems have
HI column densities 10^13.7<N(HI)<10^15.6 cm^-2. At least ~1/3 of the
individual OVI sub-systems have temperatures T<1x10^5 K and cannot originate in
collisionally ionized gas. Photoionization by a hard UV background field
reproduces well the ionic ratios for metallicities in the range 10^-2.5 -
10^-0.5 solar, with possibly sub-solar N/C relative abundance. For [O/C]=0, the
sizes inferred for the OVI clouds are in some cases larger than the maximum
extent implied by the Hubble flow. This constraint is fulfilled assuming a
moderate overabundance of oxygen relative to carbon. For a soft UV ionizing
spectrum, an overabundance of O/C is required, [O/C]~0.0-1.3. For a hard(soft)
UV spectrum and [O/C]=0(1), the OVI regions have overdensities in 10-40.
