Three solid state formation routes have been proposed in the past to explain the observed abundance of water in space: the hydrogenation reaction channels of atomic oxygen (O + H), molecular oxygen (O(2) + H), and ozone (O(3) + H). New data are presented here for the third scheme with a focus on the reactions O(3) + H, OH + H and OH + H(2), which were difficult to quantify in previous studies. A comprehensive set of H/D-atom addition experiments is presented for astronomically relevant temperatures. Starting from the hydrogenation/deuteration of solid O(3) ice, we find experimental evidence for H(2)O/D(2)O (and H(2)O(2)/D(2)O(2)) ice formation using reflection absorption infrared spectroscopy. The temperature and H/D-atom flux dependence are studied and this provides information on the mobility of ozone within the ice and possible isotope effects in the reaction scheme. The experiments show that the O(3) + H channel takes place through stages that interact with the O and O(2) hydrogenation reaction schemes. It is also found that the reaction OH + H(2) (OH + H), as an intermediate step, plays a prominent (less efficient) role. The main conclusion is that solid O(3) hydrogenation offers a potential reaction channel for the formation of water in space. Moreover, the nondetection of solid ozone in dense molecular clouds is consistent with the astrophysical picture in which O(3) + H is an efficient process under interstellar conditions.