A precise description of the injective envelope of a spatial continuous trace
C*-algebra A over a Stonean space Delta is given. The description is based on
the notion of a weakly continuous Hilbert bundle, which we show to be a
Kaplansky--Hilbert module over the abelian AW*-algebra C(Delta). We then use
the description of the injective envelope of A to study the first- and
second-order local multiplier algebras of A. In particular, we show that the
second-order local multiplier algebra of A is precisely the injective envelope
of A.