Context. The evolution of a circumstellar disk from its gas-rich protoplanetary stage to its gas-poor debris stage is not understood well. It is apparent that disk clearing progresses from the inside-out on a short time scale and models of photoevaporation are frequently used to explain this. However, the photoevaporation rates predicted by recent models differ by up to two orders of magnitude, resulting in uncertain time scales for the final stages of disk clearing. Aims. Photoevaporation theories predict that the final stages of disk-clearing progress in objects that have ceased accretion but still posses considerable material at radii far from the star. Weak-line T Tauri stars (WTTS) with infrared emission in excess of what is expected from the stellar photosphere are likely in this configuration. We aim to provide observational constraints on theories of disk-clearing by measuring the dust masses and CO content of a sample of young (1.8−26.3 Myr) WTTS. Methods. We used ALMA Band 6 to obtain continuum and 12 CO(2−1) line fluxes for a sample of 24 WTTS stars with known infrared excess. For these WTTS, we inferred the dust mass from the continuum observations and derived disk luminosities and ages to allow comparison with previously detected WTTS. Results. We detect continuum emission in only four of 24 WTTS, and no 12 CO(2−1) emission in any of them. For those WTTS where no continuum was detected, their ages and derived upper limits suggest they are debris disks, which makes them some of the youngest debris disks known. Of those where continuum was detected, three are possible photoevaporating disks, although the lack of CO detection suggests a severely reduced gas-to-dust ratio. Conclusions. The low fraction of continuum detections implies that, once accretion onto the star stops, the clearing of the majority of dust progresses very rapidly. Most WTTS with infrared excess are likely not in transition but are instead young debris disks, whose dust is either primordial and has survived disk-clearing, or is of second-generation origin. In the latter case, the presence of giant planets within these WTTS might be the cause.