Class III stars are those in star forming regions without large non-photospheric infrared emission, suggesting recent dispersal of their protoplanetary disks. We observed 30 class III stars in the 1-3 Myr Lupus region with ALMA at ${sim}856mu$m, resulting in 4 detections that we attribute to circumstellar dust. Inferred dust masses are $0.036{-}0.093M_oplus$, ${sim}1$ order of magnitude lower than any previous measurements; one disk is resolved with radius ${sim}80$ au. Two class II sources in the field of view were also detected, and 11 other sources, consistent with sub-mm galaxy number counts. Stacking non-detections yields a marginal detection with mean dust mass ${sim}0.0048M_oplus$. We searched for gas emission from the CO J=3-2 line, and present its detection to NO Lup inferring a gas mass ($4.9 {pm} 1.1$) ${times}10^{-5} M_oplus$ and gas-to-dust ratio $1.0{pm}0.4$. Combining our survey with class II sources shows a gap in the disk mass distribution from $0.09{-}2M_oplus$ for ${>}0.7M_odot$ Lupus stars, evidence of rapid dispersal of mm-sized dust from protoplanetary disks. The class III disk mass distribution is consistent with a population model of planetesimal belts that go on to replenish the debris disks seen around main sequence stars. This suggests that planetesimal belt formation does not require long-lived protoplanetary disks, i.e., planetesimals form within ${sim}$2 Myr. While all 4 class III disks are consistent with collisional replenishment, for two the gas and/or mid-IR emission could indicate primordial circumstellar material in the final stages of protoplanetary disk dispersal. Two class III stars without sub-mm detections exhibit hot emission that could arise from ongoing planet formation processes inside ${sim}1$ au.
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