  Let $E/F$ be an elliptic curve defined over a number field $F$ with complex
multiplication by the ring of integers of an imaginary quadratic field $K$ such
that the torsion points of $E$ generate over $F$ an abelian extension of $K$.
In this paper we prove the $p$-part of the Birch and Swinnerton-Dyer formula
for $E/F$ in analytic rank $1$ for primes $p>3$ split in $K$. This was
previously known for $F=\mathbb{Q}$ by work of Rubin as a consequence of his
proof of the Mazur--Swinnerton-Dyer ``main conjecture'' for rational CM
elliptic curves, but the problem remained wide open for general $F$.
  The approach in this paper, based on a novel application of an idea of
Bertolini--Darmon--Prasanna to consider a carefully chosen decomposable
Rankin--Selberg convolution of two CM modular forms having the Hecke
$L$-function of interest as one of the factors, circumvents the use of $p$-adic
heights and Bertrand's $p$-adic transcendence results in previous approaches.
It also yields a proof of similar results for CM abelian varieties $A/K$, and
for CM modular forms of higher weight.
