  Let E/F be a finite Galois extension of totally real number fields and let p
be a prime. The `p-adic Stark conjecture at s=1' relates the leading terms at
s=1 of p-adic Artin L-functions to those of the complex Artin L-functions
attached to E/F. We prove this conjecture unconditionally when E/Q is abelian.
We also show that for certain non-abelian extensions E/F the p-adic Stark
conjecture at s=1 is implied by Leopoldt's conjecture for E at p. Moreover, we
prove that for a fixed prime p, the p-adic Stark conjecture at s=1 for E/F
implies Stark's conjecture at s=1 for E/F. This leads to a `prime-by-prime'
descent theorem for the `equivariant Tamagawa number conjecture' (ETNC) for
Tate motives at s=1. As an application of these results, we provide strong new
evidence for special cases of the ETNC for Tate motives and the closely related
`leading term conjectures' at s=0 and s=1.
