Hyperfine-structure transitions in the ground state of the hydrogen-like muonium atom (mu+e-) have been induced and observed for the first time in vacuo. The 12S1/2,F= 0-1 2S1/2, F = 1 interval was measured in zero magnetic field by microwave spectroscopy. The transition frequency of 4463 310(25) kHz is in agreement with theoretical calculations and with values which were obtained earlier in noble-gas atmospheres. Future precision zero-field measurements render the possibility of testing bound-state Quantum ElectroDynamics (QED) at a very high level of precision and may reveal contributions from new physics beyond the standard model.