  Let $C(B,N)$ be the free associative conformal algebra generated by a set $B$
with a bounded locality $N$. Let $S$ be a subset of $C(B,N)$. A
Composition-Diamond lemma for associative conformal algebras is firstly
established by Bokut, Fong, and Ke in 2004 \cite{BFK04} which claims that if
(i) $S$ is a Gr\"obner-Shirshov basis in $C(B,N)$, then (ii) the set of
$S$-irreducible words is a linear basis of the quotient conformal algebra
$C(B,N|S)$, but not conversely. In this paper, by introducing some new
definitions of normal $S$-words, compositions and compositions to be trivial,
we give a new Composition-Diamond lemma for associative conformal algebras
which makes the conditions (i) and (ii) equivalent. We show that for each ideal
$I$ of $C(B,N)$, $I$ has a unique reduced Gr\"obner-Shirshov basis. As
applications, we show that Loop Virasoro Lie conformal algebra and Loop
Heisenberg-Virasoro Lie conformal algebra are embeddable into their universal
enveloping associative conformal algebras.
