[en] Muonium (M or μ⁺e^-) to antimuonium (bar M or μ^-e⁺) conversion is not allowed in the standard theory of electroweak interactions in which the lepton number of each lepton generation is separately conserved. Several extensions of the standard mode, however, permit the M → bar M conversion. Extensions of the standard model with a doubly charged Higgs triplet, for example, allows G_MbarM, the coupling constant characterizing the strength of the interaction leading to the conversion, to be as large as 10G_F, where G_F is the Fermi coupling constant. A new experimental search for the spontaneous M → bar M conversion, conducted at the Clinton P. Anderson Meson Physics Facility (LAMPF), is described in this thesis. A separated low momentum μ⁺ beam of average intensity 3 x 10^5/3 was used to produce M atoms in vacuum by the beam-foil method. The signature of an bar M event was a triple coincidence among two cascade muonic X-ray photons from the capture of the μ^- from bar M by a high-Z target and a secondary electron emitted from the target upon the impact of bar M. The photons were detected by a large solid-angle NaI(Tl) detector which had good resolution in energy, time and position. The secondary electrons were electrostatically collected and guided on to a microchannel plate detector (MCP). Observation of the M → bar M conversion would indicate that lepton family number is not a separately, additively conserved quantum number and provide a useful probe of new physics beyond the standard model. No bar M events in excess of background were observed and the 90% confidence level upper limit of G_MbarM is measured to be 6.9G_F which improves the best previous result by a factor of three