  We show that the number of copies of ${\Bbb Q}_p/{\Bbb Z}_p$ in the
Tate-Shafarevich group of an elliptic curve $E$ over ${\Bbb Q}$ with complex
multipication, is at most $2p - g$, where $g$ is the rank of $E({\Bbb Q})$, and
for all sufficiently large good ordinary primes $p$.
