A process for manufacturing a carbon fiber or graphite fiber includes melt-spinning a carbonaceous pitch, infusibilizing the resulting pitch fiber, carbonizing, and when necessary graphitizing the fiber. A precursor of the carbonaceous pitch is centrifuged in a molten state to separate the precursor pitch simultaneously into a solids slurry layer, an optically isotropic phase, and an optically anisotropic phase, and the latter is utilized as the carbonaceous pitch.