A potentially new kind of glass transition may exist among phase-separated regions in . We have observed a very large damping of ultrasonic waves in the dynamically phase-segregated state of this material. This damping is connected to the motion of structural interfaces at megahertz frequencies, which is a much faster time scale than that of magnetic relaxation effects. At lower temperatures, the dynamics of the phase-separated state freeze at the glass transition. Our observations link the onset of this freezing to a sudden decrease in mobility of interfaces between structurally dissimilar phase-segregated regions.