Rabi oscillations in the V(15) single molecule magnet embedded in the surfactant (CH(3))(2)[CH(3)(CH(2))(16)CH(2)](2)N(+) have been studied at different microwave powers. An intense damping peak is observed when the Rabi frequency Ω(R) falls in the vicinity of the Larmor frequency of protons ω(N). The experiments are interpreted by a model showing that the damping (or Rabi) time τ(R) is directly associated with decoherence caused by electron-nuclear cross relaxation in the rotating reference frame. This decoherence induces energy dissipation in the range ω(N) - σ(e) < Ω(R) < ω(N), where σ(e) is the mean superhyperfine field induced by protons at V(15). Weaker decoherence without dissipation takes place outside this window. Specific estimations suggest that this rapid cross relaxation in a resonant microwave field, observed for the first time in V(15), should also take place, e.g., in Fe(8) and Mn(12).