  For every elliptic curve $E$ which has complex multiplication (CM) and is
defined over a number field $F$ containing the CM field $K$, we prove that the
family of $p^{\infty}$-division fields of $E$, with $p \in \mathbb{N}$ prime,
becomes linearly disjoint over $F$ after removing an explicit finite subfamily
of fields. We then give a necessary condition for this finite subfamily to be
entangled over $F$, which is always met when $F = K$. In this case, and under
the further assumption that the elliptic curve $E$ is obtained as a base-change
from $\mathbb{Q}$, we describe in detail the entanglement in the family of
division fields of $E$.
