A Two Micron All Sky Survey view of the Sagittarius dwarf galaxy. III. Constraints on the flattening of the galactic halo

M giants selected from the TwoMicron All Sky Survey (2MASS) have been used to trace streams of tidal debris apparently associated with the Sagittarius dwarf spheroidal galaxy (Sgr) that entirely encircle the Galaxy. While the Sgr M giants are generally aligned with a single great circle on the sky, we measure a difference of 10.degrees4 +/- 2.degrees6 between the mean orbital poles of the great circles that best fit debris leading and trailing Sgr, which can be attributed to the precession of Sgr's orbit over the range of phases explored by the data set. Simulations of the destruction of Sgr in potentials containing bulge, disk, and halo components best reproduce this level of precession along the same range of orbital phases if the potential contours of the halo are only slightly flattened, with the ratio of the axis length perpendicular to and in the disk in the range q = 0.90-0.95 (corresponding to isodensity contours with q(rho) similar to 0.83-0.92). Oblate halos are strongly preferred over prolate (q(rho)> 1) halos, and flattenings in the potential of q less than or equal to 0.85 (q(rho) less than or equal to 0.75) and q greater than or equal to 1.05 (q(rho) greater than or equal to 1.1) are ruled out at the 3 sigma level. More extreme values of q less than or equal to 0.80 (q(rho) greater than or equal to 0.6) and q greater than or equal to 1.25 (q(rho) greater than or equal to 1.6) are ruled out at the 7 and 5 sigma levels, respectively. These constraints will improve as debris with larger separation in orbital phases is found.