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$^{8−9}$M$_ \odot$ 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 \sim$0.7. The majority of clumpy disks in our sample have a high probability of containing AGN. Their [Oiii]$\lambda$5007 emission line is strongly excited, inconsistent with low-metallicity star formation alone. [Neiii]$\lambda$3869 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 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$^{−1}$, BH growth rates $\dot m$$_{BH}$$\sim$ 10$^{−2}$M$_\odot$yr$^{−1}$ , and that these AGN are substantially obscured in X-rays. This moderate-luminosity mode could provide a large fraction of today's BH mass with a high duty cycle (>10%), accretion bursts with higher luminosities being possible over shorter phases. Violent instabilities at high redshift (giant clumps) are a much more efficient driver of BH growth than the weak instabilities in nearby spirals (bars), and the evolution of disk instabilities with mass and redshift could explain the simultaneous downsizing of star formation and of BH growth.