The disk around AU Microscopii, an M1 star in the β Pictoris moving group, is extraordinarily well suited for comparison with the β Pic debris disk (type A5 V). We use far-UV absorption spectroscopy of AU Mie to probe its edge-on disk for small amounts of H2, the primary constituent of gas giant planets. Our conservative upper limit on the line-of-sight H2 column density is 1.7 × 1019 cm-2, which is 18.5 times lower than the limit obtained recently from nondetection of submillimeter CO emission by Liu et al. In addition, there is a hint of H2 absorption at a column density an order of magnitude or more below our upper limit. The H2-to-dust ratio in the AU Mic disk is less than 6:1, similar to that in the β Pic disk. This shows that the primordial gas has largely been dissipated in less than about 12 Myr for both disks, despite their very different stellar masses and luminosities. It is extremely difficult to form a giant planet around AU Mic with current core accretion scenarios in such a short time. © 2005. The American Astronomical Society. All rights reserved.