Fourth-Order Electrodynamic Corrections to the Lamb Shift

The fourth-order radiative correction to the slope at q(2) = 0 of the Dirac form factor of the free-electron vertex Is calculated using computer techniques. The result m(2)partial derivative F(1)((4))(0)/partial derivative q(2) = (alpha/pi)(2) (0.48 +/- 0.07) disagrees with previous calculations and implies a new theoretical value for the order alpha(2) (Z alpha)(4)mc(2) contribution to the Lamb shift. The new values for the 2S(1/2) - 2P(1/2) separation in H and D are increased by 0. 35 +/- 0. 07 MHz and are in good agreement with the results of recent experiments.