Let
A
A
be a semisimple Banach algebra and
M
(
A
)
M(A)
the algebra of double multipliers of
A
A
. We show that
M
(
A
)
M(A)
is isomorphic to
(
A
∗
∗
,
∘
)
({A^{ * * }}, \circ )
if and only if
A
A
has the following properties: (1)
A
A
is Arens regular, (2)
A
A
has a weak approximate identity, and (3)
π
(
A
)
\pi (A)
is an ideal of
(
A
∗
∗
,
∘
)
({A^{ * * }}, \circ )
.