We have studied the rare weak radiative hyperon decay Xi degrees -->Sigma degrees gamma in the KTeV experiment at Fermilab. We have identified 4045 signal events over a background of 804 events. The dominant Xi degrees -->Lambdapi degrees decay, which was used for normalization, is the only important background source. An analysis of the acceptance of both modes yields a branching ratio of B(Xi degrees -->Sigma degrees gamma)/B(Xi degrees -->Lambdapi degrees ) = (3.34+/-0.05+/-0.09)x10(-3). By analyzing the final state decay distributions, we have also determined that the Sigma degrees emission asymmetry parameter for this decay is alpha(XiSigma) = -0.63+/-0.09.