For a locally compact group
G
G
let
L
1
(
G
,
ω
λ
)
{L^1}(G,\omega \lambda )
be a weighted group algebra. We characterize compact and weakly compact multipliers on
L
1
(
G
,
ω
λ
)
{L^1}(G,\omega \lambda )
. This characterization is employed to find a necessary and sufficient condition for
L
1
(
G
,
ω
λ
)
{L^1}(G,\omega \lambda )
to be an ideal in its second dual space, where the second dual is equipped with an Arens product. In the special case where
ω
(
t
)
=
1
(
t
∈
G
)
\omega (t) = 1(t \in G)
, we deduce a result due to K. P. Wong that if
G
G
is a compact group, then
L
1
(
G
,
λ
)
{L^1}(G,\lambda )
is an ideal in its second dual space and its converse due to S. Watanabe.