This was one of the first two great successes of quantum electrodynamics, the other one being the Lamb shift.
In from freeman Dyson Web of Stories interview (1998) Dyson mentions that the original key experiment was from Kusch and Foley from Columbia University, and that in 1948, Julian Schwinger reached the correct value from his calculations.
Published on Physical Review by Polykarp Kusch and Foley.