Let
A
A
be a weak-
∗
*
Dirichlet algebra in
L
∞
(
m
)
{L^\infty }(m)
and let
H
∞
(
m
)
{H^\infty }(m)
be the weak-
∗
*
closure of
A
A
in
L
∞
(
m
)
{L^\infty }(m)
. It may happen that there are minimal weak-
∗
*
closed subalgebras of
L
∞
(
m
)
{L^\infty }(m)
that contain
H
∞
(
m
)
{H^\infty }(m)
properly. In this paper it is shown that if there is a minimal, proper, weak-
∗
*
closed superalgebra of
H
∞
(
m
)
{H^\infty }(m)
, then, in fact, that algebra is the unique least element in the lattice of all proper weak-
∗
*
closed superalgebras of
H
∞
(
m
)
{H^\infty }(m)
.