in the 1970's stark made precise conjectures about the leading term of the taylor series at s=0 for artin l-functions. in the rank one setting when the order vanishing is exactly one, these conjectures relate the derivative of the l-function at s=0 to the logarithm of a unit in an abelian extension of the base field. in this talk, we will define a p-adic l-function and state a p-adic stark conjecture in the rank one setting when the base field is a quadratic field. we prove our conjecture in the case when the base field is imaginary quadratic and the prime p is split, and discuss numerical evidence in the other cases.