The method of Doppler-free two-photon laser spectroscopy is an ideal m
ethod for studying transitions between the 1S and 2S states in one-ele
ctron systems. An experiment at the Rutherford Appleton Laboratory has
observed an unambiguous signal in the exotic muonium (mu+e-) atom. Th
e transition frequency is DELTAnu1S-2S = 2455529002(33)(46) MHz, where
the first error is of a statistical nature and the second one arises
from systematic corrections, which mainly are related to the propertie
s of the high-power pulsed laser system. There is agreement with a pri
or, less accurate, independent experiment at KEK and with QED theory w
ithin two standard deviations. The Lamb shift contributions are tested
at the 8 x 10(-3) level. The muonium-hydrogen and muonium-deuterium i
sotope shifts of the 1S-2S transition have a high potential for a prec
ise mass determination of the positive muon.