The structure of water around methane during hydrate crystallization from aqueous solutions of methane is studied using neutron diffraction with isotopic substitution over the temperature range 18 degrees C to 4 degrees C, and at two pressures, 14.5 and 3.4 MPa. The carbon-oxygen pair correlation functions, derived from empirical potential structure refinement of the data, indicate that the hydration sphere around methane in the liquid changes dramatically only once hydrate has formed, with the water shell around methane being about 1 Angstrom larger in diameter in the crystal than in the liquid. The methane coordination number in the liquid is around 16 +/- 1 water molecules during hydrate formation, which is significantly smaller than the value of 21 +/- 1 water molecules found for the case when hydrate is fully formed. Once hydrate starts to form, the hydration shell around methane becomes marginally less ordered compared to that in the solution above the hydrate formation temperature. This suggests that the hydration cage around methane in the liquid may be different from that when hydrate is forming and from that found in the hydrate crystal structure. Methane-methane radial distribution functions show that methane molecules can adopt a range of separations during hydrate formation, corresponding to the more distorted nature of the methane-water correlations. There is noticeable ordering of the methane molecules with a monolayer of water molecules between them once hydrate has formed. The dipole moments of the hydrating water molecules lie mostly tangential to the methane-water axis, both before, during, and after hydrate formation.