  We give a precise definition and produce a path-integral computation of the
normalized partition function of the abelian U(1) Chern-Simons field theory
defined in a general closed oriented 3-manifold. We use the Deligne-Beilinson
formalism, we sum over the inequivalent U(1) principal bundles over the
manifold and, for each bundle, we integrate over the gauge orbits of the
associated connection 1- forms. The result of the functional integration is
compared with the abelian U(1) Reshetikhin-Turaev surgery invariant.
