  The decay asymmetries of the weak radiative Hyperon decays Xi0 \to Lambda
gamma and Xi0 \to Sigma0 gamma have been measured with high precision using
data of the NA48/1 experiment at CERN. From about 52000 Xi0 \to Lambda gamma
and 15000 Xi0 \to Sigma0 gamma decays, we obtain for the decay asymmetries
alpha(Xi0 \to Lambda gamma) = -0.704 +/- 0.019(stat) +/- 0.064(syst) and
alpha(Xi0 \to Sigma0 gamma) = -0.729 +/- 0.030(stat) +/- 0.076(syst),
respectively. These results are in good agreement with previous experiments,
but more precise.
