X-ray photons scattered by the interstellar medium carry information about dust distribution, dust grain model, scattering cross section, and the distance of the source; they also take a longer time than unscattered photons to reach the observer. Using a cross-correlation method, we study the light curves of the X-ray dust scattering halo of Cyg X-1, observed with the Chandra X-ray Observatory. Significant time lags are found between the light curves of the point source and its halo. This time lag increases with the angular distance from Cyg X-1, implying a dust concentration at a distance along the line of sight (LOS) of 2.0 kpc x (0.876 +/- 0.002) from the Earth. By fitting the observed light curves of the halo at different radii with simulated light curves, we obtain a width of Delta L = 33(-13)(+18) pc of this dust concentration. The origin of this dust concentration is still not clearly known. The advantage of our method is that we need no assumption of scattering cross section, dust grain model, or dust distribution along the LOS. Combining the derived dust distribution from the cross-correlation study with the surface brightness distribution of the halo, we conclude that the two commonly accepted models of dust grain size distribution need to be modified significantly.