We have studied the rare weak radiative hyperon decay Xi degrees --> Sigma
degrees gamma in the KTeV experiment at Fermilab. We have identified 4045 s
ignal events over a background of 804 events. The dominant Xi degrees --> L
ambda pi degrees decay, which was used for normalization, is the only impor
tant background source. An analysis of the acceptance of both modes yields
a branching ratio of B(Xi degrees --> Sigma degrees gamma) / B(Xi degrees -
-> Lambda pi degrees) = (3.34 +/- 0.05 +/- 0.09) X 10(-3). By analyzing the
final state decay distributions, we have also determined that the Sigma de
grees emission asymmetry parameter for this decay is alpha (Xi Sigma) = -0.
63 +/- 0.09.