High-Velocity Clouds (HVCs) have radial velocities that cannot be explained
by the global Galactic rotation; their distances remain mostly unknown, an
d their true nature and origin are still a mystery. Some of them could be o
f galactic origin, or belong to tidal streams drawn by the Milky-Way/Magell
anic Clouds interaction, or could even be intergalactic clouds infalling on
to the Local Group. In the latter hypothesis, they play a major role in the
hierarchical formation scenario of the Milky-Way and are connected to the
Lyman-limit absorption systems. In any case, the determination of their phy
sical state (density, temperature, internal structure, abundances, excitati
on) will help to discriminate between current theories on their origin and
nature. A recent UV measurement (Richter et al. 1999) has discovered for th
e first time in a HVC the molecular phase that was previously searched for
without success, through CO emission. Previous non detections could be due
either to metallicity problems, or insufficient excitation (because of low
density). Low-excitation molecular gas may, however, be detectable through
absorption. Here we report on a sensitive search for HCO+(1-0) absorption l
ines in front of 27 quasars, already known to be strong millimetric continu
um sources. Except for one tentative case, no detection was obtained in mos
t HVCs, although HCO+(1-0) was clearly detected towards galactic low-veloci
ty clouds. We discuss the implications of this result.