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

Faint disks around classical T Tauri stars: small but dense enough to form planets?