We present a dynamical mean-field model for molecular motions in a supercooled polymer melt. A macromolecule is represented by a harmonic chain undergoing Brownian motion whose bead mobilities fluctuate between zero and a finite value. These fluctuations mimic the dynamic obstacles formed by the chain segments surrounding a given segment,;whose effects become more pronounced as T decreases. The rate of these mobility fluctuations is determined self-consistently by equating it to the asymptotic long-time;relaxation rate of the shortest-wavelength Rouse mode. The resulting fluctuating rate vanishes as c, the equilibrium fraction of mobile beads, approaches a threshold value c*. As c-->c*, relaxation times become;arbitrarily large, permitting the modeling of fluids as T approaches T-g. Calculations of autocorrelation functions of Rouse mode coordinates and of segmental mean-squared displacements are presented: and compared to results from recent simulations of melts at low temperatures. The deviations from the Rouse model observed in the simulations are features of this theory.