We present the details of a powerful new technique used in an exact calculation of the contribution of 6 sixth-order vertex graphs to the anomalous magnetic moment of the electron. By introducing four-dimensional spherical coordinates, the momentum-space integrals can be evaluated directly. The angular integrations areperformed using some simple properties of the Gegenbauer polynomials. For these graphs, the remaining convergent radial integrands are simple rational functions which are readily integrated. The divergent integrals are treated separately to extract all terms which do not vanish in the infrared and ultraviolet limits. The actual processing of the many terms involved was done using the algebraic

W e compute a precise value for three more graphs contributing to the g factor o f the electron in sixth order. After comparing with other numerical and analytic evaluations, we give an updated "best" theoretical estimate of the g factor, and compare it to the experimental value.

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Five graphs contributing to g -2 of the electron in sixth order are evaluated to high precision. With these results, the dominant error in the theoretical prediction for g -2 arises from the eighth-order uncertainty. Updated comparisons of theory and experiment are given. The techniques employed are briefly characterized.