Are super-radiant phase transitions possible?

An argument is given which rules out super-radiant phase transitions in which the atomic states involved are connected by electric-dipole matrix elements. The argument is based on gauge invariance, and is valid even in the presence of direct interatomic interactions. Its relevance is demonstrated by presenting a number of two-level models exhibiting super-radiant phase transitions which are not forbidden by previous sum-rule arguments. As a preliminary to the argument, the nature of the super-radiant phase transition is discussed, and a simple model is given which reproduces its main features. It is argued, finally, that in at least one case in which our argument fails a super-radiant phase transition is possible in principle.