  We study the disordered Heisenberg spin chain, which exhibits many body
localization at strong disorder, in the weak to moderate disorder regime. A
continued fraction calculation of dynamical correlations is devised, using a
variational extrapolation of recurrents. Good convergence for the infinite
chain limit is shown. We find that the local spin correlations decay at long
times as $C \sim t^{-\beta}$, while the conductivity exhibits a low frequency
power law $\sigma \sim \omega^{\alpha}$. The exponents depict sub-diffusive
behavior $ \beta < 1/2, \alpha> 0 $ at all finite disorders, and convergence to
the scaling result, $\alpha+2\beta = 1$, at large disorders.
