We discuss a refinement of Rubin's integral version of Stark's Conjecture for abelian $L$-functions of arbitrary order of vanishing at $s=0$. This generalizes the $v$-adic refinement of the abelian, order of vanishing $1$ integral Stark Conjecture formulated by Gross in the 1980s, and predicts a link between special values of derivatives of $p$-adic and global $L$--functions. Time permitting, we will also show how our refinement relates to a recent strengthening of Gross's Conjecture due to Tate.