A search with the large solid angle spectrometer (ARES) for the lepton-number-nonconserving decay mu+ --> e+e+e- has been carried out. Cylindrical multiwire proportional chambers and cylindrical scintillation hodoscopes were used. The upper limit for the branching ratio GAMMA (mu+ --> e+e+e-) / GAMMA(mu+ --> e+nu-e-nu-mu) less-than-or-equal-to 3.6 x 10(-11) (90% C.L.) has been derived. The investigation has been performed in the Laboratory of Nuclear Problems, JINR.