  Masur showed that a Teichmuller geodesic that is recurrent in the moduli
space of closed Riemann surfaces is necessarily determined by a quadratic
differential with a uniquely ergodic vertical foliation. In this paper, we show
that a divergent Teichmuller geodesic satisfying a certain slow rate of
divergence is also necessarily determined by a quadratic differential with
unique ergodic vertical foliation. As an application, we sketch a proof of a
complete characterization of the set of nonergodic directions in any double
cover of the flat torus branched over two points.
