In two previous papers with Yi-Jen Lee, we defined and computed a notion of Reidemeister torsion for the Morse theory of closed 1-forms on a finite dimensional manifold. The present paper gives an a priori proof that this Morse theory invariant is a topological in- variant. It is hoped that this will provide a model for possible generalizations to Floer theory. 2000 Mathematics Subject Classification: 57R70, 57Q10; 37C27, 57R58. In two papers with Yi-Jen Lee (HL1, HL2), we defined a notion of Reidemeister torsion for the Morse theory of closed 1-forms on a finite dimensional manifold. We consider the flow dual to the 1-form via an auxiliary metric. Our invariant, which we call I, multiplies the algebraic Reidemeister torsion of the Novikov complex, which counts flow lines between critical points, by a zeta function which counts closed orbits of the flow. For a closed 1-form in a real multiple of an integral cohomology class, i.e. d of a circle-valued function, we proved in the above papers that I equals a form of topological Reidemeister torsion due to Turaev. This implies a posteriori that I is invariant under homotopy of the circle-valued function and the auxiliary metric. In this paper we reprove these results using an opposite approach: we first prove a priori that I is a topological invariant, depending only on the cohomology class of the closed 1-form. We then deduce that I agrees with Turaev torsion, by using invariance to reduce to the easier case of an exact 1-form. This approach has two advantages. First, it works for closed 1-forms in an arbitrary cohomology class, thus extending the results of our previous papers. Second, and perhaps more importantly, the proof of invariance here should provide a model for the possible construction of torsion in Floer theory. The contents of this paper are as follows. In §1 we review the definition of the in- variant I, state the main results, and outline the proofs. In §2 and §3 we prove that I is invariant. The strategy is to study how I varies in a generic one parameter family of 1-forms and metrics. In §2, we prepare for this analysis by classifying the bifurcations that generically occur, and we also deal with the complication that infinitely many bifurcations may occur in a finite time. The heart of the paper is in §3, where we an- alyze what happens in each individual bifurcation. While the torsion of the Novikov