  We examine the radiative decays of the Lambda (1405) to octet baryons, Lambda
+ gamma and Sigma + gamma. In the limit of SU(3) symmetry the decay rates for
these two modes are related, and we compute the leading correction to the
relation in chiral perturbation theory. Such measurements will allow the sign
of the strong coupling $g_{\Lambda^*}$ to be determined. The SU(3) violating
radiative decay to the baryon decuplet Sigma* + gamma dominated by one-loop
graphs is also discussed. Experiments planned for the Jefferson Laboratory
should be able to measure some, if not all, of the processes considered.
