  Context. Radiation-driven mass loss is key to our understanding of
massive-star evolution. However, for low-luminosity O-type stars there are big
discrepancies between theoretically predicted and empirically derived mass-loss
rates (called the weak-wind problem). Aims. We compute radiation-line-driven
wind models of a typical weak-wind star to determine its temperature structure
and the corresponding impact on ultra-violet (UV) line formation. Methods. We
carried out hydrodynamic simulations of the line-deshadowing instability (LDI)
for a weak-wind star in the Galaxy. Subsequently, we used this LDI model as
input in a short-characteristics radiative transfer code to compute synthetic
UV line profiles. Results. We find that the line-driven weak wind is
significantly shock heated to high temperatures and is unable to cool down
effciently. This results in a complex temperature structure where more than
half of the wind volume has temperatures significantly higher than the stellar
effective temperature. Therefore, a substantial portion of the weak wind will
be more ionised, resulting in a reduction of the UV line opacity and therefore
in weaker line profiles for a given mass-loss rate. Quantifying this, we find
that weak-wind mass-loss rates derived from unsaturated UV lines could be
underestimated by a factor of between 10 and 100 if the high-temperature gas is
not properly taken into account in the spectroscopic analysis. This offers a
tentative basic explanation for the weak-wind problem: line-driven weak winds
are not really weaker than theoretically expected, but rather a large portion
of their wind volume is much hotter than the stellar effective temperature.
