In this paper, we will construct the p-adic zeta function for a
non-commutative p-extension of a totally real number field such that the finite
part of its Galois group is a p-group with exponent p. We first calculate the
Whitehead groups of the Iwasawa algebra and its canonical Ore localization by
using Oliver-Taylor's theory on integral logarithms. Then we reduce the main
conjecture to certain congruences among abelian p-adic zeta pseudomeasures
constructed by Deligne-Ribet and Serre. Finally we prove these congruences by
using the theory of Deligne-Ribet and certain induction.