  The Fredrickson-Andersen (FA) model---a kinetically constrained lattice
model---displays an ergodic to non-ergodic transition with a slow two-step
relaxation of dynamical correlation functions close to the transition point. We
derive an asymptotically exact solution for the dynamical occupation
correlation function of the FA model on the Bethe lattice by identifying an
exact expression for its memory kernel. The exact solution fulfills a scaling
relation between critical exponents and allows to calculate the exponents
explicitly. In addition, we propose an approximate dynamics that describes
numerical data away from the critical point over many decades in time.
