  In this paper we show that in the presence of an anti-symmetric tensor
$B$-background, Witten's star algebra for open string fields persists to
possess the structure of a direct product of commuting Moyal pairs. The
interplay between the noncommutativity due to three-string overlap and that due
to the $B$-background is our main concern. In each pair of noncommutative
directions parallel to the $B$-background, the Moyal pairs mix string modes in
the two directions and are labeled, in addition to a continuous parameter, by
{\it two} discrete values as well. However, the Moyal parameters are
$B$-dependent only for discrete pairs. We have also demonstrated the large-$B$
contraction of the star algebra, with one of the discrete Moyal pairs dropping
out while the other giving rise to the center-of-mass noncommutative function
algebra.
