We present a new experimental value for the magnetic moment of the electron bound in hydrogenlike carbon (12C5+): g(exp) = 2.001 041 596 (5). This is the most precise determination of an atomic g(J) factor so far. The experiment was carried out on a single 12C5+ ion stored in a Penning trap. The high accuracy was made possible by spatially separating the induction of spin flips and the analysis of the spin direction. The current theoretical value amounts to g(th) = 2.001 041 591 (7). Together experiment and theory test the bound-state QED contributions to the g(J) factor of a bound electron to a precision of 1%.