  We discuss the non-perturbative renormalization group flow of Quantum
Electrodynamics (QED) coupled to Quantum Einstein Gravity (QEG) and explore the
possibilities for defining its continuum limit at a fixed point that would lead
to a non-trivial, i.e. interacting field theory. We find two fixed points
suitable for the Asymptotic Safety construction. In the first case, the
fine-structure constant vanishes at the fixed point and its infrared
("renormalized") value is a free parameter not determined by the theory itself.
In the second case, the fixed point value of the fine-structure constant is
non-zero, and its infrared value is a computable prediction of the theory.
