The question of the nature of water's glass transition has continued to be disputed over many years. Here we use slow heating scans (0.4 K min^{-1}) of compact amorphous solid water deposited at 77 K and an analysis of the accompanying changes in the small-angle neutron scattering signal, to study mesoscale changes in the ice network topology. From the data we infer the onset of rotational diffusion at 115 K, a sudden switchover from nondiffusive motion and enthalpy relaxation of the network at <121 K to diffusive motion across sample grains and sudden pore collapse at >121 K, in excellent agreement with the glass transition onset deduced from heat capacity and dielectric measurements. This indicates that water's glass transition is linked with long-range transport of water molecules on the time scale of minutes and, thus, clarifies its nature. Furthermore, the slow heating rates combined with the high crystallization resistance of the amorphous sample allow us to identify the glass transition end point at 136 K, which is well separated from the crystallization onset at 144 K-in contrast to all earlier experiments in the field.