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The program of quantizing the gravitational field with the help of affinefield variables is continued. For completeness, a review of the selectioncriteria that singles out the affine fields, the alternative treatment ofconstraints, and the choice of the initial (before imposition of theconstraints) ultralocal representation of the field operators is initiallypresented. As analogous examples demonstrate, the introduction and enforcementof the gravitational constraints will cause sufficient changes in the operatorrepresentations so that all vestiges of the initial ultralocal field operatorrepresentation disappear. To achieve this introduction and enforcement of theconstraints, a well characterized phase space functional integralrepresentation for the reproducing kernel of a suitably regularized physicalHilbert space is developed and extensively analyzed.Comment: LaTeX, 42 pages, no figure

Noncanonical quantization of gravity. II. Constraints and the physical Hilbert space