The results of an experimental search with a new method for the spontaneous conversion of muonium to antimuonium are reported. The upper limit for G/sub M//sub M-bar/, the coupling constant characterizing the strength of the interaction leading to the conversion, is measured to be G/sub M//sub M-bar/<7.5G/sub F/ (90% confidence level), where G/sub F/ is the Fermi coupling constant. This result is about a factor of 3 lower than the previous limit and begins to probe predictions of the left-right-symmetric theory with a doubly charged Higgs triplet.