We study numerically the long-time dynamics of a one-dimensional Bose-Einstein condensate expanding in a speckle or impurity disorder potential. Using the mean-field Gross-Pitaevskii equation, we demonstrate subdiffusive spreading of the condensate for long times. We find that interaction-assisted hopping between normal modes leads to this subdiffusion. A possible (partial) reason why the root-mean-square (rms) width saturates in the experiment [ Nature (London) 453 891 (2008)] is provided. We suggest that observing both the participation length and the rms width of a condensate, rather than only the rms width, could provide a more complete description of the long-time behavior of ultracold atoms in disorder potentials. Our study confirms subdiffusive spreading in spatially continuous disordered interacting models and highlights new features which spatially discrete models do not possess.