This paper introduces a new approach to measure the muon magnetic moment anomaly a(mu) = (g - 2)/2 and the muon electric dipole moment (EDM) d(mu) at the J-PARC muon facility. The goal of our experiment is to measure a(mu) and d(mu) using an independent method with a factor of 10 lower muon momentum, and a factor of 20 smaller diameter storage-ring solenoid compared with previous and ongoing muon g - 2 experiments with unprecedented quality of the storage magnetic field. Additional significant differences from the present experimental method include a factor of 1000 smaller transverse emittance of the muon beam (reaccelerated thermal muon beam), its efficient vertical injection into the solenoid, and tracking each decay positron from muon decay to obtain its momentum vector. The precision goal for a(mu) is a statistical uncertainty of 450 parts per billion (ppb), similar to the present experimental uncertainty, and a systematic uncertainty less than 70 ppb. The goal for EDM is a sensitivity of 1.5 x 10(-21) e.cm.