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Let E be an elliptic curve over Q with complex multiplication by the ring ofintegers of an imaginary quadratic field K. In 1991, by studying a certainspecial value of the Katz two-variable p-adic L-function lying outside therange of $p$-adic interpolation, K. Rubin formulated a p-adic variant of theBirch and Swinnerton-Dyer conjecture when $E(K)$ is infinite, and he provedthat his conjecture is true for E(K) of rank one.  When E(K) is finite, however, the statement of Rubin's original conjecture nolonger applies, and the relevant special value of the appropriate $p$-adicL-function is equal to zero. In this paper we extend our earlier work and givean unconditional proof of an analogue of Rubin's conjecture when E(K) isfinite.

On Rubin's variant of the p-adic Birch and Swinnerton-Dyer conjecture II