In 1993, Sczech defined an $n-1$ cocycle on $GL_n(Z)$ valued in a certain space of distributions. He showed that specializations of this cocyle yield the values of the partial zeta functions of totally real fields of degree $n$ at nonpositive integers. In this talk, I will describe an integral refinement of Sczech's cocycle. By introducing a "smoothing" prime $l$, we define an $n-1$ cocycle on a congruence subgroup of $GL_n(Z)$ valued in a space of $p$-adic measures. We prove that the specializations analogous to those considered by Sczech produce the $p$-adic L-functions of totally real fields. We also consider certain other specializations that conjecturally yield the Gross-Stark units defined over abelian extensions of these fields. This is joint work with Pierre Charollois.