(abridged) Most Class II sources (of nearby star forming regions) are surrounded by disks with weak millimeter continuum emission. These "faint" disks may hold clues to the disk dissipation mechanism. We attempt to determine the characteristics of such faint disks around classical T Tauri stars, and to explore the link between disk faintness and the proposed disk dispersal mechanisms (accretion, viscous spreading, photo-evaporation, planetary system formation). We performed high-angular resolution (0.3") imaging of a small sample of disks (9 sources) with low 1.3mm continuum flux (mostly <30 mJy) with the IRAM Plateau de Bure interferometer and simultaneously searched for 13CO (or CO) J=2-1 line emission. Using a simple parametric disk model, we determine characteristic sizes of the disks, in dust and gas, and we constrain surface densities in the central 50 AU. All disks are much smaller than the bright disks imaged so far, both in continuum and 13CO lines (5 detections). In continuum, half of the disks are very small, with characteristic radii less than 10AU, but still have high surface density values. Small sizes appear to be the main cause for the low disk luminosity. Direct evidence for grain growth is found for the three disks that are sufficiently resolved. Low continuum opacity is attested in two systems only, but we cannot firmly distinguish between a low gas surface density and a lower dust emissivity resulting from grain growth. We report a tentative discovery of a 20 AU radius cavity in DS Tau, bringing the proportion of transitional disks to a similar value to that of brighter sources, but cavities cannot explain the low mm flux. This study highlights a category of very compact dust disks, still exhibiting high surface densities, which may represent up to 25 % of the whole disk population, but its origin is unclear with the current data alone.< Réduire