  Given an abelian, CM extension K of any totally real number field k, we
consider two conjectures `of Stark type'. The `Integrality Conjecture' concerns
the image of a p-adic map `\mathfrak{s}_{K/k,S}' determined by the minus-part
of the S-truncated equivariant L-function for K/k at s=1. It is connected to
the Equivariant Tamagawa Number Conjecture of Burns and Flach. The `Congruence
Conjecture' says that \mathfrak{s}_{K/k,S} gives an explicit reciprocity law
for the element predicted by the corresponding Rubin-Stark Conjecture for
K^+/k. We study the general properties of these conjectures and prove one or
both of them under various hypotheses, notably when p does not divide [K:k],
when k=Q or when K is absolutely abelian.
