The muon anomalous magnetic moment has been measured in a new experiment at Brookhaven. Polarized muons were stored in a superferric ring, and the angular frequency difference, omega(a), between the spin precession and orbital frequencies was determined by measuring the time distribution of high-energy decay positrons. The ratio R of omega(a) to the Larmor precession frequency of free protons, omega(p), in the storage-ring magnetic field was measured. We find R = 3.707 220(48) X 10(-3). With mu(mu)/mu(p) = 3.183 345 47(47) this gives a(mu+) = 1 165 925(15) X 10(-9) (+/-13 ppm), in good agreement with the previous CERN measurements for mu(+) and mu(-) and of approximately the same precision.