  The nature of the ordering of the three-dimensional isotropic Heisenberg spin
glass with nearest-neighbor random Gaussian coupling is studied by extensive
Monte Carlo simulations. Several independent physical quantities are measured
both for the spin and for the chirality, including the correlation-length
ratio, the Binder ratio, the glass order parameter, the overlap distribution
function and the non-self-averageness parameter. By controlling the effect of
the correction-to-scaling, we have obtained a numerical evidence for the
occurrence of successive chiral-glass and spin-glass transitions at nonzero
temperatures, T_{CG} > T_{SG} > 0. Hence, the spin and the chirality are
decoupled in the ordering of the model. The chiral-glass exponents are
estimated to be \nu_{CG}=1.4+-0.2 and \eta_{CG}=0.6+-0.2, indicating that the
chiral-glass transition lies in a universality class different from that of the
Ising spin glass. The possibility that the spin and chiral sectors undergo a
simultaneous Kosterlitz-Thouless-type transition is ruled out. The chiral-glass
state turns out to be non-self-averaging, possibly accompanying a one-step-like
peculiar replica-symmetry breaking. Implications to the chirality scenario of
experimental spin-glass transitions are discussed.
