  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 ${\sim}856\mu$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.
