  Let E be an elliptic curve over Q with complex multiplication by the ring of
integers of an imaginary quadratic field K. In 1991, by studying a certain
special value of the Katz two-variable p-adic L-function lying outside the
range of $p$-adic interpolation, K. Rubin formulated a p-adic variant of the
Birch and Swinnerton-Dyer conjecture when $E(K)$ is infinite, and he proved
that his conjecture is true for E(K) of rank one.
  When E(K) is finite, however, the statement of Rubin's original conjecture no
longer applies, and the relevant special value of the appropriate $p$-adic
L-function is equal to zero. In this paper we extend our earlier work and give
an unconditional proof of an analogue of Rubin's conjecture when E(K) is
finite.
