Ionization processes of a water dimer have been investigated by means of full dimensional direct ab-initio trajectory method. The structure of (H2O)(2) before the ionization was simulated at 10 K up to 2 ps by means of a direct ab-initio molecular dynamics (MD) method. Sixty geometrical configurations were selected from the MD calculation, and then the dynamics of (H2O)(2)(+) were calculated by means of a direct ab-initio trajectory method under a constant energy condition. The trajectories on two electronic state potential energy surfaces of (H2O)(2)(+), i.e., the (2)A" and 2A' states, were full-dimensionally calculated under the C-s symmetry. For the (2)A" state, a complex composed of H3O+ and OH, which is expressed by (H3O+)OH, was only formed as a long-lived complex. On the other hand, in the ionization to the (2)A' state, two reaction channels, the complex formation (H3O+)OH ((2)A') and direct dissociation (H3O+ + OH) were open as product channels. The dynamics calculations indicated that the proton of H2O+ transferred to H2O within 100-150 fs for the (2)A" state, whereas the proton transfer at the (2)A' state occurs faster (30-50 fs) than that of the (2)A" state. The mechanism of the ionization dynamics of (H2O)(2) was discussed on the basis of theoretical results.