Strongly coupled positronium, considered in its pseudoscalar sector, is studied in the framework of relativistic quantum constraint dynamics. Case's method of self-adjoint extension of singular potentials, which avoids explicit introduction of regularization cut-offs, is adopted. It is found that, as the coupling constant \alpha increases, the bound state spectrum undergoes an abrupt change at the critical value \alpha = \alpha_c = 1/2. For \alpha > \alpha_c, the mass spectrum displays, in

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... ion to the existing states for \alpha < \alpha_c, a new set of an infinite number of bound states concentrated in a narrow band starting at mass W=0. In the limit \alpha going to \alpha_c from above, these states shrink to a single massless state with a mass gap with the rest of the spectrum. This state has the required properties to represent a Goldstone boson and to signal a spontaneous breakdown of chiral symmetry. It is suggested that the critical coupling constant \alpha_c be viewed as a possible candidate for an ultra- violet stable fixed point of QED, with a distinction between two phases, joined to each other by a first-order chiral phase transition.