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(Abridged) We present results from a JCMT/SCUBA 850 um search for cold dustaround nearby young stars belonging to the beta Pic (t~12 Myr) and the LocalAssociation (t~50 Myr) moving groups. Unlike most past sub-mm studies, oursample was chosen on the basis of stellar age. Our observations achieve aboutan order of magnitude greater sensitivity in dust mass compared to previouswork in this age range. We detected two of the three M dwarfs in our sample at850 um, GJ 182 and GJ 803. GJ 182 may also possess a 25 um excess, indicativeof warm dust in the inner few AU of its disk. For GJ 803 (AU Mic), sub-mmmapping finds that the 850 um emission is unresolved. A non-detection of the CO3-2 line indicates the system is gas-poor, and the SED suggests the presence ofa large inner disk hole (~17 AU = 1.7 arcsec in radius). These are possibleindications that planets at large separations can form around M dwarfs within\~10 Myr. In a companion paper (Kalas, Liu & Matthews 2004), we confirm theexistence of a dust disk around GJ 803 using optical coronagraphic imaging.Given its youthfulness, proximity, and detectability, the GJ 803 disk will be avaluable system for studying disk, and perhaps planet, formation in greatdetail. Overall, sub-mm measurements of debris disks point to a drop in dustmass by a factor of about 10^3 within the first ~10 Myr, with the subsequentdecline in the masses of sub-mm detected disks consistent with t^{-0.5} tot^{-1}.Comment: 9 pages, ApJ, in press. Minor changes made to reflect final published  manuscrip

A Submillimeter Search of Nearby Young Stars for Cold Dust: Discovery of Debris Disks around Two Low‐Mass Stars