  Improving earlier estimates of several authors we show that the number E(X)
of Goldbach exceptional even integers (that is, even integers which cannot be
written as the sum of two primesw) below X satisfies tho bound E(X) < X^0.72
for sufficiently large X. In the proof crucial role is played by the explicit
formula for the contribution of the major arcs proved in Part I of this series
(arXiv:1804.05561). In this new version some misprints are corrected and an
explanatory sentence is added to the common proof of theorems C and D.
