  We provide evidence for a correlation between the presence of giant clumps
and the occurrence of active galactic nuclei (AGN) in disk galaxies. Giant
clumps of 10^8-9 Msun arise from violent gravitational instability in gas-rich
galaxies, and it has been proposed that this instability could feed
supermassive black holes (BH). We use emission line diagnostics to compare a
sample of 14 clumpy (unstable) disks and a sample of 13 smoother (stable) disks
at redshift z~0.7. The majority of clumpy disks in our sample have a high
probability of containing AGN. Their [OIII] emission line is strongly excited,
inconsistent with low-metallicity star formation alone. [NeIII] excitation is
also higher. Stable disks rarely have such properties. Stacking ultra sensitive
Chandra observations (4 Ms) reveals an X-ray excess in clumpy galaxies, which
cannot be solely due to star formation and confirms the presence of AGN. The
clumpy galaxies in our intermediate-redshift sample have properties typical of
gas-rich disk galaxies rather than mergers, being in particular on the Main
Sequence of star formation. This suggests that our findings apply to the
physically-similar and numerous gas-rich unstable disks at z>1. Using the
observed [OIII] and X-ray luminosities, we conservatively estimate that AGN
hosted by clumpy disks have typical bolometric luminosities of the order of a
few 10^43 erg/s, BH growth rates ~10^-2 Msun/yr, and that these AGN are
substantially obscured in X-rays. This moderate-luminosity mode could be
sufficient to provide a large fraction of today's BH mass over a couple of Gyr
given that our observations suggest a high duty cycle (>10%), accretion bursts
with higher luminosities being possible over shorter phases. The observed
evolution of disk instabilities with mass and redshift could explain the
simultaneous downsizing of star formation and of BH growth.
