  In the first part of the paper we describe the dual \ell^2(A)^{\prime} of the
standard Hilbert C*-module \ell^2(A) over an arbitrary (not necessarily unital)
C*-algebra A. When A is a von Neumann algebra, this enables us to construct
explicitly a self-dual Hilbert A-module \ell^2_{\text{strong}}(A) that is
isometrically isomorphic to \ell^2(A)^{\prime}, which contains \ell^2(A), and
whose A-valued inner product extends the original inner product on \ell^2(A).
This serves as a concrete realization of a general construction for Hilbert
C*-modules over von Neumann algebras introduced by W. Paschke.
  Then we introduce a concept of a weak Bessel sequence and a weak frame in
Hilbert C*-modules over von Neumann algebras. The dual \ell^2(A)^{\prime} is
recognized as a suitable target space for the analysis operator. We describe
fundamental properties of weak frames such as the correspondence with
surjective adjointable operators, the canonical dual, the reconstruction
formula, etc; first for self-dual modules and then, working in the dual, for
general modules.
  In the last part of the paper we describe a class of Hilbert C*-modules over
L^{\infty}(I), where I is a bounded interval on the real line, that appear
naturally in connection with Gabor (i.e. Weyl-Heisenberg) systems. We then
demonstrate that Gabor Bessel systems and Gabor frames in L^2(\Bbb R) are in a
bijective correspondence with weak Bessel systems and weak frames of translates
by a in these modules over L^{\infty}[0,1/b], where a,b>0 are the lattice
parameters. In this setting some well known results on Gabor systems are
discussed and some new are obtained.
