  We demonstrate agreement between measurements and ab initio calculations of
the frequency shifts caused by distributed cavity phase variations in the
microwave cavity of a primary atomic fountain clock. Experimental verification
of the finite element models of the cavities gives the first quantitative
evaluation of this leading uncertainty and allows it to be reduced to delta nu
/ nu = 8.4\times10^-17. Applying these experimental techniques to clocks with
improved microwave cavities will yield negligible distributed cavity phase
uncertainties, less than \pm1\times10^-17.
