We report results of a sensitive search for cold dust and molecular gas in the disks around 8 T Tauri stars in the high-latitude cloud MBM 12. Interferometric observations of 3 mm continuum emission in 5 fields containing 6 of the objects, and literature values for the remaining two, limit the disk masses to M_disk < 0.04 - 0.09 M_sun (gas+dust), for a gas:dust mass ratio of 100 and a distance of 275 pc. By coadding the 3 mm data of our five fields, we set an upper limit to the average disk mass of <M_disk>(N=5) < 0.03 M_sun. Simultaneous observation of the CS J=2-1 and the N2H+ 1-0 lines show no emission. Single-dish observations of the 13CO 2-1 line limit the disk mass to (5 - 10) * 10^-4 M_sun for a standard CO abundance of 2 * 10^-4. Depletion of CO by up to two orders of magnitude, through freezing out or photodissociation, can reconcile these limits. These mass limits lie within the range found in the Taurus-Auriga and rho Oph star-forming regions (0.001 - 0.3 M_sun), and preclude conclusions about possible decrease in disk mass over the 1 - 2 Myr age range spanned by the latter two regions and MBM 12. Our observations can exclude the presence in MBM 12 of T Tauri stars with relatively bright and massive disks such as T Tau, DG Tau, and GG Tau.
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