  An all high-latitude sky survey for cool carbon giant stars in the Galactic
halo has revealed 75 such stars, of which the majority are new detections. Of
these, more than half are clustered on a Great Circle on the sky which
intersects the center of Sagittarius dwarf galaxy (Sgr) and is parallel to its
proper motion vector, while many of the remainder are outlying Magellanic Cloud
C-stars. A pole-count analysis of the carbon star distribution clearly
indicates that the Great Circle stream we have isolated is statistically
significant, being a 5-6 sigma over-density. These two arguments strongly
support our conclusion that a large fraction of the Halo carbon stars
originated in Sgr. The stream orbits the Galaxy between the present location of
Sgr, 16 kpc from the Galactic center, and the most distant stream carbon star,
at ~60 kpc. It follows neither a polar nor a Galactic plane orbit, so that a
large range in both Galactic R and z distances are probed. That the stream is
observed as a Great Circle indicates that the Galaxy does not exert a
significant torque upon the stream, so the Galactic potential must be nearly
spherical in the regions probed by the stream. We present N-body experiments
simulating this disruption process as a function of the distribution of mass in
the Galactic halo. A likelihood analysis shows that, in the Galactocentric
distance range 16 kpc < R < 60 kpc, the dark halo is most likely almost
spherical. We rule out, at high confidence levels, the possibility that the
Halo is significantly oblate, with isodensity contours of aspect q_m < 0.7.
This result is quite unexpected and contests currently popular galaxy formation
models. (Abridged)
